Trilinear Model Research Articles

The Constitutive Relationship for Corroded Steel Bars: Model and Analysis.

Combining the theoretical derivation and numerical calculations, the characteristic changes of the tensile constitutive relation of corroded steel bars and their underlying mechanisms are studied. Corroded steel bars are regarded as a combination of three parts, which include uncorroded part, corroded part with variable cross section, and uniform corroded part. It is assumed that in all three parts the steel material follows a simplified trilinear constitutive relation of a mild steel material (elasticity, yielding, and hardening), from which an analytical model describing the overall tensile constitutive relation of the corroded steel bar is developed. Based on the experimental data of slotted steel bars, the validation of the present analytical model is provided. The results show that the trilinear model can give relatively accurate prediction of the characteristic parameters of corroded steel bars. The influences of corrosion rate on the mechanical properties of corroded steel bars are examined using the proposed model.

Tensile properties of wire and fibre reinforced cementitious matrix composites for strengthening of masonry

Currently, wire and fibre reinforced inorganic matrices are receiving a lot of attention amongst the structural engineers, especially for strengthening of existing masonry structures. The motivation behind this development stems from the drawbacks of organic (epoxy resin) matrices of fibre reinforced polymers (FRP). A number of cement-based composites have been developed in the recent years in form of fibre reinforced cementitious matrix (FRCM) and wire reinforced cementitious matrix (WRCM) composites. Since the development of composites made of cementitious matrix is still under development, the evaluation of mechanical properties of such systems is a crucial step in their application. The tensile strength of the composite is an important parameter for the design of the strengthening system. This article presents an experimental study on 75 welded wire mesh (WWM) and basalt fibre mesh (BFM) reinforcement and corresponding composite specimens. The specimens are tested following the relevant ASTM and ACI standards. The mechanical behaviour of the composite is compared with the constituent reinforcement. The failure modes and tensile behaviour of the composites before and after cracking are also discussed. The test results are analyzed and compared with ‘Aveston-Cooper-Kelly (ACK)’ theory and the ‘simplified tri-linear (STL) model’.

Trilinear Model for the Out-of-Plane Seismic Assessment of Vertically Spanning Unreinforced Masonry Walls

AbstractOut-of-plane failure of masonry walls is often responsible for the partial collapse of unreinforced masonry structures. Modeling the out-of-plane response of these walls is therefore key in...

Probabilistic Characteristics of Moment Capacity and Rotational Stiffness of Wedge Joints Used in Support Systems Reflecting Reused Members

The moment capacity and rotational stiffness of wedge joints, which connect vertical and horizontal members of assembled support systems, were evaluated experimentally considering the characteristics of reused members. Since temporary structures, such as supports, tend to be reused, experiments were conducted with reused members, and the normality of the measured data was assessed. The lower and upper limits of the 95% confidence intervals of the moment capacity and rotational stiffness of wedge joints with reused members were determined. Experiments were also conducted on a joint system with new members to analyze the influence of reused members. In integrating both new and reused members, the maximum moments of wedge joints were observed to be normally distributed. The lower limit of the 95% confidence interval of the maximum moment of joints was 0.997 kNm, and the upper limit was 1.074 kNm. The rotational stiffness of the wedge joint was evaluated using a trilinear model. The initial rotational stiffness decreased with continued use of the joint. The average rotational stiffness of the joint, analyzed by combining the results for new and reused members, was found to be 22.475 kNm/rad for the first interval, 4.705 kNm/rad for the second interval, and 1.577 kNm/rad for the third interval. The lower limit of the 95% confidence interval of the initial rotational stiffness was 20.688 kNm/rad, and the upper limit was 24.262 kNm/rad.

Load Transfer of Offshore Open-Ended Pipe Piles Considering the Effect of Soil Plugging

Open-ended pipe piles have been increasingly used as the foundations for offshore structures. Considering the soil plugging effect, a novel analytical model is proposed in this paper to study the load transfer mechanism of open-ended pipe piles. A trilinear model for the external shaft friction was introduced, while a rigid plastic model was adopted to describe the load transfer at the pile-plug interface. Furthermore, an equilibrium equation of the soil plug was proposed, based on the hypothesis of a trilinear distribution of lateral earth pressure. The pile end resistance was analyzed by dividing it into two parts, i.e., the soil plug and pile annulus, the behaviors of which were described by the double broken line model. A calculation example was carried out to analyze the load transfer properties of the open-ended pipe piles. As a validation, similar load transfer processes of the open-ended pile were also captured in a newly built discrete element method model, mimicking the 100g centrifuge testing conditions.

Characterization on tensile behaviors of fracture process zone of nuclear graphite using a hybrid numerical and experimental approach

This paper reports an experimental and numerical study on the fracture properties of nuclear graphite NG-CT-01 produced in China. Three-point bending tests were performed on center-notch beams and electronic speckle pattern interferometry (ESPI) technique was employed to measure the full-field deformation of the beams. Besides, finite element analysis was performed by using extended finite element method to simulate the fracture behavior of graphite. Numerical results were compared with the experimental results and excellent agreement was obtained, validating the reliability and accuracy of the numerical model. Based on the strain field measured by ESPI technique, the formation and evolution of the fracture process zone (FPZ) in graphite were observed. The tension softening curve (TSC) of the graphite was determined by using a hybrid numerical and experimental approach. The influences of initial crack length to beam depth ratio on the fracture responses of graphite beam, including the load-displacement curves, double-K fracture parameters, TSCs and the length of the FPZ, were evaluated. In order to explain the correlation between the tension softening behaviors and the fracture mechanism of the FPZ, a tri-linear model was proposed. Furthermore, double-K fracture criterion was employed to predict the crack propagation in graphite.

An improved analytical approach for simulating the lateral kinematic distress of deepwater offshore pipelines

Offshore pipelines are critical infrastructures and any possible damage may have devastating financial and environmental consequences. Earthquake-related geohazards (such as strong ground motion, active seismic faults, submarine landslides and debris flows) consist crucial threats that an offshore pipeline has to overcome. The main aim of the current study is to examine analytically a seabed-laid offshore pipeline subjected to a lateral kinematic distress due to a submarine landslide or a debris flow. Extra emphasis is given on the impact of pipe-soil interaction on the pipe response, by the realistic representation of the soil resistance via a tri-linear model. Firstly, the proposed analytical model is validated with a numerical model utilizing the finite-element method. Subsequently, various combinations of soil parameters and loading conditions that affect the examined problem are investigated with realistic input data taken from the offshore section of the high-pressure natural-gas pipeline TAP (Trans Adriatic Pipeline) in the Adriatic Sea. Finally, useful conclusions are drawn regarding the applicability and the efficiency of the proposed approach.

Seismic behavior of strengthened square reinforced concrete columns under combined loadings

This paper presents an experimental study examining the seismic behavior of square reinforced concrete (RC) columns subjected to combined loadings. In this study, nine full-scale square RC columns were made with the same structural properties. Five of them were strengthened with high-performance ferrocement laminate and bonded steel plates in order to investigate the effects of strengthening on the seismic behavior of RC columns under combined loadings. Finally, the failure mode, hysteretic load-displacement response, bearing capacity, energy dissipation, ductility and plastic hinge characteristics were recorded and analyzed. The results show that combined loadings reduced the bearing capacity and changed the failure modes and deformation characteristics. Strengthened RC columns had significantly improved bearing capacity and different failure modes under the same combined loadings in contrast to the non-strengthened RC columns. The vertical eccentricity had little effect on the mechanical properties of columns, while the horizontal eccentricity significantly weakened the mechanical properties of columns. The mechanical properties of uniaxial compression-bending-shear-torsion columns were the worst. The tri-linear model calculation parameters under different failure modes and combined loadings were fixed and validated by experimental results.

Comparison of Bi- and Tri-Linear PLS Models for Variable Selection in Metabolomic Time-Series Experiments.

Metabolomic studies with a time-series design are widely used for discovery and validation of biomarkers. In such studies, changes of metabolic profiles over time under different conditions (e.g., control and intervention) are compared, and metabolites responding differently between the conditions are identified as putative biomarkers. To incorporate time-series information into the variable (biomarker) selection in partial least squares regression (PLS) models, we created PLS models with different combinations of bilinear/trilinear X and group/time response dummy Y. In total, five PLS models were evaluated on two real datasets, and also on simulated datasets with varying characteristics (number of subjects, number of variables, inter-individual variability, intra-individual variability and number of time points). Variables showing specific temporal patterns observed visually and determined statistically were labelled as discriminating variables. Bootstrapped-VIP scores were calculated for variable selection and the variable selection performance of five PLS models were assessed based on their capacity to correctly select the discriminating variables. The results showed that the bilinear PLS model with group × time response as dummy Y provided the highest recall (true positive rate) of 83–95% with high precision, independent of most characteristics of the datasets. Trilinear PLS models tend to select a small number of variables with high precision but relatively high false negative rate (lower power). They are also less affected by the noise compared to bilinear PLS models. In datasets with high inter-individual variability, bilinear PLS models tend to provide higher recall while trilinear models tend to provide higher precision. Overall, we recommend bilinear PLS with group x time response Y for variable selection applications in metabolomics intervention time series studies.

An analytical model for analyzing the impact of fracturing fluid-induced formation damage on rate transient behavior in tight formations

An enormous amount of fracturing fluid is irreducible in the tight formation after hydraulic fracturing stimulation. This will cause serious damage to the fracture face as well as the matrix close to the hydraulic fractures by invading into the matrix. Modeling the effect of water blockage on rate transient behavior is significant in reservoir engineering. This paper presents a modified trilinear model to analyze the rate transient behavior after flowback period. Single-phase flow is assumed in this stage because most injected water is hard to be reduced. For damage to the matrix caused by rock-fluid interaction, clay swelling, and polymer adsorption, a fracturing fluid invasion layer is added to the model between hydraulic fractures and the inner stimulated reservoir volume to capture this kind of damage. A skin factor is used to capture the damage to the matrix-fracture face caused by gel filter cake formation and polymer adsorption. The analytical model is benchmarked using the commercial numerical simulator Eclipse. Several synthetic cases and a field case are studied, the results show that water blockage has a great impact on the productivity of the well and decline curves. The effect of choking effect, convergence flow, and gel filter cake could be incorporated using a skin factor in the model. In addition, two more flow regimes could be observed on the theoretical type curve if the well is affected by water blockage, and the two flow regimes would happen before linear flow dominated by the inner stimulated zone. Besides, the duration of the inner zone linear flow regime could be shortened if the permeability of the fracturing fluid invasion layer is lower and invasion depth is larger. These phenomena can be used in matching the field production data from water blockage affected wells, which exhibit a transition flow regime before the formation linear flow regime. This transition flow regime could be matched quite better using the proposed model than the classical trilinear flow model. The reason is that the linear flow regime could be much longer if the water blockage effect is not considered in the model. On the other hand, the skin factor has a different impact on the flow regimes, so the water blockage effect could be characterized properly only by adding a skin factor to the model. The main contribution of this paper is the provision of an analytical model for evaluating the formation damage caused by irreducible fracturing fluid, analyzing production data and making forecasts.

The Analytical Approach to Evaluate the Load‐Displacement Relationship of Rock Bolts

Fully grouted rock bolts are widely used in civil engineering and mining engineering, playing a significant role in keeping the stability and safety of excavations. In this paper, the load transfer mechanism of fully grouted rock bolts was studied with an analytical model. A trilinear model was used to depict the bond‐slip behaviour of the bolt/grout interface. The displacement of the confining medium was involved in this analytical model. Then, the shear stress propagation along the bolt/grout interface was analysed in the elastic, elastic‐softening, elastic‐softening‐debonding, softening‐debonding, and debonding stages. Experimental pull‐out tests were used to validate this analytical model. There was a good correlation between experimental and analytical results. A parametric study was conducted to evaluate the influence of Young’s modulus of the confining medium, the shear strength of the bolt/grout interface, and the residual shear strength of the bolt/grout interface on the load transfer performance of rock bolts. The results show that increasing Young’s modulus of the confining medium was beneficial for improving the load transfer performance of rock bolts. However, once Young’s modulus of the confining medium was beyond a critical limit, it had marginal effect on the peak load of rock bolts. Furthermore, increasing the shear strength of the bolt/grout interface and the residual shear strength of the bolt/grout interface led to rising of the peak load of rock bolts. However, compared with the residual shear strength of the bolt/grout interface, increasing the shear strength of the bolt/grout interface had more apparent effect in improving the peak load of rock bolts.

Investigation on True Stress‐Strain Curves of Flat and Corner Regions of Cold‐Formed Section Using 3D Digital Image Correlation Method

The true stress‐strain curve is the critical method to describe the practical material mechanical performance and the essential precondition to develop the advanced numerical simulation. Experimental, analytical, and numerical procedures were performed in present research to investigate the true stress‐strain curves of flat and corner regions of the cold‐formed channel section. The coupon tests with the 3D digital image correlation system were conducted on flat and corner specimens to directly obtain the true stress‐strain curves. The experimental results indicate that the tensile secondary‐hardening phenomenon at the plastic strain stage was observed in the true stress‐strain curves of flat coupons, and initial strain hardening behavior was produced in that of corner coupons. Flat region exhibits a significant improvement of true ultimate strength compared to the engineering value. The stress status of the corner region is developed to ultimate strength at the early strain phase and exhibits a slight increase compared with the nominal values at the plastic phase. Cold‐rolling action limits the ductility performance of the corner region, which highly restrains the tensile strain hardening at the plastic condition. Thus, the true yielding strength of the corner region is obviously higher, but the true ultimate strength is significantly lower than that of the flat region. Together with the optical measuring results, a trilinear model with two‐stage strain hardening and a simplified trilinear models were established for describing the true stress‐strain curves of flat and corner regions, respectively. The load‐displacement curves from numerical simulations fit very well with those of coupon tests, which validate the reliability of the optic measurement and the dependability of the simplified constitutive models.

Effects of mechanical properties of adhesive and CFRP on the bond behavior in CFRP-strengthened steel structures

The bond between carbon fiber-reinforced polymer (CFRP) composites and steel is a key issue in CFRP-strengthened steel structures. The determination of key parameters in the bond strength model remains debatable. We studied the bond behavior between CFRP and steel, focusing on the effects of different types of epoxy adhesive and CFRP on the failure modes, bond–slip relationships, and bond strength parameters. The results show that due to the use of various materials, some specimens fail in interface debonding characterized by brittle failure, whereas others have cohesive failure or CFRP delamination characterized by ductile failure. The bond–slip models showed gentle-slope descending branches for interfaces with ductile failure, which is missing in brittle failure. The bond–slip relationships of most specimens can be simplified as bilinear models. However, trilinear models should be used for the specimens failing in the delamination and tearing of CFRP. As for the modes of interface debonding and CFRP superficial delamination, the maximum shear stresses in the interface linearly correlate with the elastic moduli of adhesive. The nonlinear mechanical behavior of interfaces can be well simulated by a cohesive zone model.

Behavior analysis of FRP tube/filling strengthened steel members under axial compression

Abstract Influence of the ultimate strength of stee This study presents a new fiber reinforced polymer tube/filling strengthening method (i.e., FRP tube/filling strengthening) that creates a new composite compression member with a novel mechanical behavior, “bilinear behavior.” Previous experimental and theoretical studies have been conducted based on the ultimate state of the section. To thoroughly study the stiffness and entire loading process of this novel member, finite element analysis (FEA) models are built in this paper. First, the model development process considers the initial defections and interface properties, and these models are verified by the experimental results. Two models of typical load-axial displacement curves describe the experimental results: a Bilinear model and a Trilinear model. Second, finite element modeling (FEM) is used to conduct a parametrical analysis that considers 7 parameters, and the results identify the mechanism of the strengthened member. Specifically, the Bilinear model corresponds to an unreinforced or relatively weakly reinforced specimen, and the Trilinear model corresponds to a relatively strongly reinforced specimen. If the exterior restraint system is weak, then the core steel buckles and subsequently yields; this behavior is described by the Bilinear model. If the exterior restraint system is strong (e.g., the length or the bending stiffness of the exterior restraint system is great), then the specimen can continue to bear a higher force after steel yielding until elastic-plastic buckling occurs; this behavior is described by the Trilinear model. Third, the FEA results and previous experimental and theoretical studies are used to obtain the stiffness and load-axial displacement curves, which are helpful for the design process. In general, relatively accurate FEA models are built in this paper that are capable of describing the mechanical behaviors of the novel member.

Two-dimensional DOA estimation for generalized coprime planar arrays: a fast-convergence trilinear decomposition approach

In this paper, we investigate the problem of two-dimensional (2D) direction of arrival (DOA) estimation of multiple signals for generalized coprime planar arrays consisting of two rectangular uniform planar subarrays. We propose a fast-convergence trilinear decomposition approach, which uses propagator method (PM) as the initialization of the angle estimation to speed the convergence of trilinear decomposition. The received signal of each subarray can be fitted into a trilinear model or parallel factor (PARAFAC) model so that the trilinear alternating least square algorithm can be used to estimate the angle information. Meanwhile, the necessary initialization of DOA estimates can be achieved via PM, which endows the proposed approach a fast convergence and subsequently results in a low complexity. Specifically, we eliminate the ambiguous estimates by utilizing the coprime property and the true DOA estimates can be achieved by selecting the nearest ones of all DOA estimates. The proposed approach can obtain the same estimation performance as the conventional PARAFAC algorithm, but with a low computational cost. Numerical simulation results are provided to validate the effectiveness and superiority of the proposed algorithm.

Experimental study on the seismic behaviour of RC beams with standing and hanging walls

Recently, earthquake damage to non-structural walls has become one of the important issues in Japan. Some buildings were demolished after the 2011 Tohoku Earthquake due to damage of non-structural walls without any significant damage in structural members. After that, several projects were launched to develop a new method to take into account the effect of non-structural walls (hanging, standing, and wing walls). In this paper, experimental test results for beam-column joints with non-structural walls are presented. The objectives of the tests were to investigate the equivalent length and hinge location of beams with hanging and standing walls. The results showed that the yield hinge located at the surface of the wing walls and beam-column joint should be modelled as rigid to estimate the deformation of the beams, regardless of the thickness and height of the wall. A tri-linear modelling method for beams with hanging and standing walls was also proposed, and its applicability was confirmed with the test results.

Hysteretic behaviour of steel fibre RC coupled shear walls under cyclic loads: Experimental study and modelling

This paper presents the hysteretic behaviour of three 1/3-scale three-storey steel fibre reinforced concrete (SFRC) coupled shear walls (CSWs) under cyclic loads. The deformation, ductility, energy dissipation, stiffness and crack propagation of the specimens are also discussed and analysed. The results show steel fibre improves the ductility and energy dissipation capacity, and restrains the crack propagation of the CSWs, and delays the degradation of their lateral stiffness and force. Based on the experiments, a simple trilinear model is developed to simulate the skeleton curve of lateral force–displacement of the SFRC CSWs. Through analysing several typical cycles of the hysteretic of these CSWs, the feature points of the proposed hysteretic model are defined which subsequently is used to evaluate the complete hysteretic behaviour of the CSWs. Using existing experimental data and this study, several representative experimental hysteretic cycles are compared with the proposed model. The result indicates a good agreement is reached between the model and experimental results.

A theoretical 4-stage shear model for single-lap torqued bolted-joint with clearances

In this paper, a theoretical 4-stage shear model for a single-lap torqued bolted-joint with clearance is proposed. The deformation mechanism of the bolted-joint indicates that the contact status is the key factor for the nonlinearity of the load-displacement curve. On the basis of the contact status, the expression of this model and its parameters are proposed. Compared with the previous Tri-linear model, this model is improved on three aspects. First, the 1st stage of the model has been improved by using the partial slip theory. Second, an additional 2nd stage is considered for taking into account the asynchronous slip of the interfaces between components and between bolt head and component. Third, the 4th stage of this model is derived from the elastic foundation beam model and the Cai-Shan Liu’s approximate contact model. Moreover, the 4-stage shear model is validated by 3-D detailed FE-models. The results show that the 4-stage model can correctly describe the shear behavior, especially the nonlinearity of the load-displacement curve in the 1st stage and the 4th stage. In addition, the effects of clearance and bolt length on the joint stiffness in each stage are presented.

Dealing with three-way data containing missing values by new weighted method for second-order calibration

Multi-way data arrays contain missing values for several reasons, such as various malfunctions of instruments, responses being outside instrument ranges, irregular measurement intervals between samples and data postprocessing. In the present study, one new method, weighted alternating penalty trilinear decomposition (W-APTLD), based on the weighted trilinear model and the idea of alternative trilinear decomposition was given to analyze three-way data arrays containing missing values. In addition, one improved core consistency diagnostic method (W-CORCONDIA) was proposed to estimate the chemical ranks of three-way data arrays containing missing values. The results of one simulation and two real data sets demonstrate that the new method W-APTLD could be used to deal with missing values and reserves the second-order advantage. When meeting excessive factors, W-APTLD could give more accurate results than weighted PARAFAC (W-PARAFAC), PARAFAC with single imputation (PARAFAC-SI) and incomplete data PARAFAC (INDAFAC). The convergence rate of W-APTLD was much faster than W-PARAFAC and PARAFAC-SI but slower than INDAFAC. Better than W-PARAFAC and PARAFAC-SI, W-APTLD could overcome the problem due to severe collinearity. In addition, this new method could be extended to analyze higher-way data arrays containing missing values.

Angle estimation and mutual coupling self-calibration for ULA-based bistatic MIMO radar

In this paper, we propose an effective scheme for angle estimation and array mutual coupling (MC) self-calibration in uniform linear arrays (ULA)-based bistatic multiple-input multiple-output (MIMO) radar. By exploiting the multidimensional inherent structure, the array data is formulated into a trilinear model. The transmit and receive direction matrices are primarily estimated via trilinear decomposition, after which the least square (LS) method is applied to obtain a rough angle estimation. Refined angles are achieved via two one-dimensional local searches. The MC coefficients are obtained via LS by utilizing the estimated angle prior. The proposed scheme can achieve automatic pairing of the estimated angles without any virtual aperture loss, thus it has better angle and MC estimation performance than existing methods. Numerical simulations verify the improvement of our scheme.