Joints connect prefabricated hollow-core slabs, the key elements of force transmission of hollow-core slab bridges. The joints are easily damaged, which affects the integrity and safety of the hollow-core slab bridge. This paper uses MIDAS FEA NX finite element analysis software to simulate the deep hinge joint segment model, comparing and analyzing the finite element simulation analysis results with the test results, and proposes the critical parameters of the hinge joint interface and concrete damage plasticity. Further, an assembled hollow slab bridge model is established to compare and analyze the force transfer performance of conventional and pinned reinforcement bridges and reveal the deep joint damage evolution process. The results showed that the hollow slab bridge damage appeared first at the hinge joint interface at the load location. Cracks in the joints can develop along the longitudinal and height directions, with the longitudinal crack length reaching 40% of the span. The vertical crack height can get the lower edge of the paving layer, increasing the distance from the load position, and the cracking height decreases symmetrically. Under an ultimate load, the hinge concrete of conventional reinforcement and pinned reinforcement hollow-core slab bridges showed significant damage in 30–70% and 40–60% of the span, respectively. Compared with the conventional reinforcement bridges, the cracking load and ultimate load of the pinned reinforcement bridges increase by 28.57% and 58.14%, respectively, and the relative deflection under 420 kN load reduces by 97.95%. The hollow slab bridges have improved the force performance and thus enhanced the integrity of the hollow-core slab bridges. Creating a solid transportation sector has achieved notable advancements, and the building of transportation infrastructure is in an advanced stage of growth as the nation quickens the pace of creating a powerful socialist modern country. As the primary setting for transportation, bridges have significant importance. Small and medium-sized bridges are the most common type of bridge construction, with many different styles. Assembled hollow-core slab bridges are one type that is frequently used in actual projects because of their advantages ,2,3, whiHowever, in the actual operation process, due to the unreasonable design and construction stage in the bridge operation process, or the increasing overload use and insufficient maintenance, the problems of bridge damage and disease continue to appear. If the joints are destroyed, the transverse connection between the slabs will be weakened , and theThe joint problem attracted the attention of several scholars who had studied the failure process and force transmission mechanism of joints. Brunesi et al.  confirmeTo enhance the mechanical behavior of hollow-core slab bridges, scholars have studied the effects of different factors on bridges. He et al.  investigWith the improvement of technology, finite element software was widely used by scholars in their research. Due to the diverse functions of finite element software, scholars used different methods to establish finite element models of hollow slab bridges. Yuan et al.  built XFIn summary, building finite element modeling of full-size hollow-core slab bridges by finite element software has been widely used by many scholars. Changing the pavement layer thickness or the joints’ form can enhance the bearing capacity of the joints and prevent premature damage. However, few scholars have studied the stress state of the old and new concrete interface of the hollow-core slab joints and the interface damage process during the whole loading process. At the same time, the cracking of the joint surface of the hinge joint is difficult to be observed directly. Moreover, the joint with pinned reinforcement has good mechanical behavior in static load tests , but fewThe detailed dimensions of the specimens with conventional reinforcement (SCM) and pinned reinforcement (SPH)  were as The concrete with a design strength of 40 MPa was used for the beam concrete, and the concrete with a design strength of 60 MPa was used for the joint concrete. Hot-rolled ribbed bars with a yield strength of 400 MPa were used for specimens. SPH was also enhanced with hinge joint reinforcement using three layers of U-shaped reinforcement in the form of pin joint reinforcement. The reinforcement details of specimens are shown in In this paper, flexural-shear loading was adopted. The oil pressure gauge was used to control the load value and loading speed during the test. The load was first loaded in increments of 5% of the predicted limit load value until it broke. Then it was continuously loaded in increments of 10% of the expected limit load value. The changes in cracks in the loading process were recorded. The test was terminated when the crack reached the lower edge of the pavement. The load at this time was called the crack penetration load.The cracking loads of SCM and SPH were 130 and 180 kN, respectively, and the crack penetration loads were 205 and 280 kN, respectively. Their damage processes were the same as shown in The finite element model, as shown in Unlike other finite element software, the reinforcement unit in Midas FEA NX used embedded truss elements, which could be directly added to the structure. This kind of element only needed to set the basic material properties of the reinforcement, and the software would automatically handle the nodes of reinforcement and concrete so that the bond slip between the reinforcement and concrete did not need to be considered. In addition, when the interface elements were set, the nodes between the precast beam concrete element and the joint concrete element could be directly transferred from connection to disconnection.In MIDAS FEA NX, the main constitutive models for simulating the mechanical behavior of concrete were the concrete smeared crack mode and the concrete plastic damage model. The concrete plastic damage model (CDP) could reasonably simulate the cracking and damage of concrete materials. This study used this model to simulate concrete plastic material properties. The specific parameter settings are shown in The constitutive reinforcement model mainly included the isotropic linear elastic, Von Mises, and fully plastic models. Among them, the Von Mises model was widely used in the analysis of metallic materials, which mainly defined the same ductile material behavior as steel and could be used as an ideal elastic-plastic material for yielding damage when the stress reached a critical value and could also define the stress-strain curve independently. Therefore, the Von Mises model was used in this paper. According to reference , the uniAccording to the static test, the surface between the joint and the beam cracked, so the discrete cracked constitutive model was used to simulate the bond surface. The essence of this model was the combination of discrete elements and interface elements, and the relative displacement relationship between elements simulated the structural cracking. When the normal stress of the interface element reached the tensile strength, the relative displacement occurred between the main elements, equivalent to the cracking of the concrete bonding surface. In this paper, the nonlinear tensile softening model was selected as the functional relationship of the intrinsic model of the interface element , and thecf is the sum of the concrete standard compressive strength, and 8 is called the average compressive strength; a = 0.025, 0.030, 0.058, corresponding to the maximum size of aggregate 8, 16, 32 mm, respectively.The material parameters of the interface element were related to the mechanical properties of the structural material and needed to be adjusted according to the test results. The material parameters of the interface element included the normal stiffness modulus, tangential stiffness modulus, tensile strength, and fracture energy. The normal stiffness modulus was equal to the elastic modulus of concrete, and the tangential stiffness modulus was equal to 0.01 times the normal stiffness modulus. The tensile strength was 0.7 times that of the concrete axial tensile strength . In the Comparing the finite element analysis results of SCM and SPH with the experimental results , it was As shown in In summary, Midas FEA NX could more accurately simulate the specimen with a joint by setting the interface element between the joint and beam. Through the comparative analysis of the results, this paper determined the interface parameters of the finite element models with different joints, and the specific values are shown in In this paper, a 10 m span hollow-core slab bridge was simulated by using Midas FEA NX. The hollow-core slab bridge adopted the bridge system of “three beams and two joints”  because Due to the extent limitation, the finite element model was presented in detail with the conventional reinforcement hollow-core slab bridge (BCM) model as an example. The finite element model of the bridge is shown in To obtain more accurate nonlinear analysis results and to ensure computational efficiency, a tetrahedral mesh of size 50 mm was used to mesh the concrete entities, as shown in There was no slip damage of the reinforcement in the statical test, so implantable truss units were used to realize the common nodal coupling between the reinforcement elements and concrete elements with a size of 20 mm. According to the different diameters of reinforcements, hollow-core slab reinforcements and joint reinforcements were divided respectively, and there were 127,397 reinforcement elements in total.2, and that of joint and pavement layer concrete was 35,992 N/mm2. Hot-rolled ribbed bars with a yield strength of 400 MPa were used for specimens, with an elastic modulus of 2 × 105 MPa and Poisson’s ratio of 0.3. The interface element material parameters of the hollow-core slab bridge are shown in The elastic modulus of beam concrete was 32,599 N/mmThe cracking height of the joint interface elements is shown in In summary, the cracking started from the joint interface at the position where the load acts. The crack developed upward with the load increase. Finally, it reached the lower edge of the pavement layer. At the same time, the cracks at the joint interface developed symmetrically along the longitudinal direction and reached 0.3 and 0.7 times the span at last, and the crack length was 4 m. The crack width was the largest at the position where the load acts, and it decreased with the increase in distance from the load.The relative deflection of the hollow-core slab on both sides of the joint is shown in According to the location of the load, the hollow-core slab close to the load was called the loading segment, and the hollow-core slab far from the load was called the unloaded segment in this paper. The stresses of the tensile reinforcement in the joints are shown in The cracking height of the joint interface elements is shown in The relative deflection of the hollow-core slab on both sides of the joint is shown in The stress trend of the tensile reinforcement in the joint is shown in The bearing capacity of the conventional reinforcement hollow-core slab bridge (BCM) and pinned reinforcement hollow-core slab bridge (BPH) are shown in For BCM, the bottom of the loading side joint interface at the load location was the first to crack when the load was 140 kN. After the cracks appeared, with the increase of load, the cracks developed along the height direction and longitudinal direction, and when the ultimate load of 680 kN was reached, the joint concrete of 0.3–0.7 times the span showed damages, and the damage range reached 4 m, which was consistent with the cracking length of the joint interface.For BPH, when the load was 180 kN, the bottom of the joint interface on the loading segment cracked at the load location, and the cracking load was increased by 28.75% compared with BCM. With the load increase, the damage to joint concrete appeared. It could be seen from Under the load, the joint damage in the mid-span was the most serious, so the tensile reinforcement stresses in the lower part of the joint in BCM and BPH were shown in The relative deflection of the hollow-core slab on both sides of the joint at the load position in mid-span is shown in The crack height and width of the joint interface for hollow-core slab bridges with different joint forms under load are shown in The crack width at the joint interface at the load position is shown in In this study, a finite element model of a 10 m span assembled hollow slab bridge was established based on the hinge joint interface and concrete plastic damage unit characteristic parameters of the finite element analysis model of deep hinge joint section specimens to numerically analyze the whole bridge to explore the hinge joint damage evolution process and provide a basis for the improvement and strengthening of the hinge joint.(1)The failure of the static load test and finite element analysis of the beam with deep joints is the cracking of the interface between the joint and the loaded segment. With the load increase, the cracking extends to the bottom of the pavement layer. According to the cracking situation, the force transmission mechanism of the joint can be divided into two stages: the uncracked stage and the cracked stage. In the uncracked stage, the joint surface mainly transfers load, and the stress of the tensile reinforcement at this position is minor. In the cracked stage, the bonding force of the joint surface at the cracking location disappears, and the bonding force between the reinforcement and concrete and the reaction force of the concrete column jointly transfer the load.(2)Compared with the results of finite element analysis and statical load test, the stress and relative deflection of reinforcement are the same, and the maximum error of load penetrating crack is 12.2%. The failure of the finite element model is mainly the cracking of the interface between the joint and the hollow-core slab. Before the crack penetration load, the joint surface presents the stress distribution of tension in the lower part and compression in the upper part, and the height of the tension zone increases gradually with the increase in load.(3)A finite element model of a 10 m span assembled hollow-core slab bridge is established using material parameters of the beam segment with deep joint to study the bridge’s force transfer performance and damage evolution. The results show that the failure of hollow-core slab bridge starts from the joint interface at the load location, and with the load increase, the damage to the joint appears, and the damage range of convention reinforcement hollow-core slab bridge under ultimate load is 3–7 m. The cracking develops along the height and longitudinal directions with the load increase, and the cracking height can reach 600 mm. The cracking length of the convention reinforcement joint and pinned reinforcement joint hollow-core slab bridge can get 4 and 2 m. The damage to the joint interface is most profound at the load location, where the interface cracking height, crack width, and relative deflection are the largest, and they decrease with the increase of distance from the load location. The stress of the tensile reinforcement in the hinge joint shows a trend of rapid growth, slow growth, and growth with the load increase.(4)Analysis of the bearing capacity, joint reinforcement stress, relative deflection, cracking height, and crack width of hollow-core slab bridges with different joints shows that the pinned reinforcement bridges have better mechanical behavior, and the ultimate load is increased by 62.79% compared with the conventional reinforcement bridges. Compared with the conventional reinforcement hollow slab bridge, when the load reached 400 kN, the stress and relative deflection of the pin-connected steel hollow slab bridge hinge joint tensioned reinforcement decreased by 96.77% and 97.58%, respectively; when the hollow slab bridge hinge joint interface cracking height developed to 600 mm, the load of the pin-connected steel hollow slab bridge increased by 34.88%; when the crack width reaches 0.01 mm, its load increases by 40.74%. Therefore, the hinge joint form of pin jointed reinforcement with groove can improve the interface bond, delay the development of hinge joint interface cracks, enhance the transverse force transfer performance of the hinge joint and the integrity of the hollow slab bridge, and then improve the force performance of the hollow slab bridge.(5)In fact, we do not find suitable results from other scholars to compare. In our subsequent study, we will continue to investigate the mechanical properties of other forms of hollow slab bridges and analyze them in comparison with the two bridges in this paper. Meanwhile, it is challenging to conduct load tests due to the large size of full-size bridges, so this study needs tests for full-size bridges. We will complete this work in a future study.


In order to improve the resource utilization of recycled concrete powder (RCP), this study aimed to investigate the effect of RCP admixture, curing age, and alkali excitation on the strength of RCP concrete. In addition, the pore structure characteristics of RCP concrete were analyzed in combination with low-field NMR. Furthermore, a gray predictive GM  model was established to predict the mechanical properties of the concrete based on the pore structure parameters, especially the compressive and flexural tensile strengths. The results of the study indicate that the mechanical properties, namely compressive strength and flexural strength, of RCP concrete exhibit an initial increase followed by a subsequent decrease with increasing RCP content at 3 d, 7 d, and 28 d curing ages. In particular, the concrete exhibits the highest mechanical properties when the RCP content reaches 10%. As the curing age increases, the RCP gradually achieves full hydration, resulting in further refinement of the concrete pores and a denser structure, which subsequently improves the mechanical properties. In addition, the strength growth rate of alkali-excited recycled concrete (ARC) showed a continuous increase, indicating that alkali excitation increasingly improved the mechanical properties of the concrete. Furthermore, the study accurately predicted the mechanical properties of RCP concrete by using GM  prediction models for its compressive strength and flexural tensile strength using pore characteristic parameters. Since the beginning of the 21st century, there has been rapid global urbanization, resulting in significant amounts of construction and demolition (C&D) waste . For exa2, Al2O3, and CaO and exhibits properties typical of supplementary cementitious materials (SCM), such as the micro-aggregate filling effect and the volcanic ash effect. Ma [2 content increases. Consequently, the compressive strength decreases as the RCP content increases.The processing of C&D waste can result in the production of recycled aggregate through a series of treatments. Although the performance of recycled aggregates may not be equal to that of natural aggregates, by improving the crushing process, surface treatment, and matrix modification, as well as adjusting the matching ratio, recycled aggregate concrete or mortar can still meet the design requirements. This has been demonstrated in previous studies ,10,11,12fect. Ma  has showfect. Ma ,19 arguefect. Ma  shows thfect. Ma ,22,23,24fect. Ma  shows thConcrete is a complex and heterogeneous multiphase material that forms numerous pores during the setting and hardening process. These pores have different morphologies, complex structures, and a wide range of sizes. At present, there is limited research on the pore structure of concrete after the addition of RCP. Most scholars ,27 have At present, there is relatively little research on the relationship between pore structure parameters and mechanical properties after the incorporation of RCP concrete. In addition, the investigation of the laws and quantitative relationship between pore structure and mechanical properties of RCP concrete under the influence of alkali excitation is particularly lacking. Therefore, this research focuses on the use of RCP derived from C&D waste in the resource recovery process. It aims to investigate the effect of replacement rate and curing age of RCP as well as alkali excitation on the mechanical properties of concrete. In addition, low-field NMR analysis is used to investigate the pore structure characteristics of the concrete. A GM  gray prediction model is then established to correlate the mechanical properties of the concrete with the pore structure parameters. The primary objective of this study is to increase the resource utilization rate of RCP while reducing the environmental pollution associated with traditional cement production. This research has strategic significance for China’s building materials’ industry to achieve the “double carbon” goal  and effe2, CaO, and Al2O3. In addition, the mineral fraction of RCP was examined using an X-ray diffractometer (XRD) and the results are shown in P.O42.5 ordinary silicate cement supplied by Jidong Cement Co., Ltd.,  was used in the test. In the experiment, ordinary river sand with a grain size ranging from 0.15 mm to 5 mm was used as the fine aggregate. The sand is properly graded and has a fineness modulus of 2.7. The coarse aggregate used was machine crushed stone with continuous grading from 4.75 mm to 26.0 mm. Detailed performance indices of the sand and coarse aggregate are shown in 2+ content, analytically pure Ca(OH)2 was used as a chemical exciter, specifically selected to stimulate the activity of the RCP , produced by the Tianjin Beichen Fangzheng Reagent Factory . A high efficiency polycarboxylic acid water reducer was used to achieve a water reduction rate of approximately 30%.In order to increase the OH concentration of the slurry and to replenish the Ca2 was selected as the activator, and the RCP was used to replace the cement at replacement rates of 10%, 20%, and 30%, respectively. The details of the mixes are listed in Based on the Technical Guidelines for Construction of Highway Cement Concrete Pavements (JTG/TF30-2014) , the oriThe procedure commenced with an initial blending of sand and coarse aggregate within a mixer for a duration of 2 min. Subsequently, cement and recycled micronized powder were introduced sequentially and mixed for 2 min to achieve homogeneity. Following this, a composite of water reducer, exciter, and test water was meticulously prepared and then introduced into the mixer for a 2-min cycle. Finally, the meticulously blended concrete was poured into molds with precision. The molded concrete specimens were then subjected to 2 min of controlled vibration on a vibrator, followed by the meticulous removal of excess mixture using a trowel, thereby ensuring strict adherence to the specified specimen dimensions.In accordance with the Test Methods of Cement and Concrete for Highway Engineering (JTG 3420-2020) , the dimIn accordance with the Test Methods of Cement and Concrete for Highway Engineering (JTG 3420-2020) , the speT2 relaxation value by the action of external gradient field, so that the water molecules in the magnetic field and the gradient field produce nuclear magnetic resonance phenomenon. According to the change of T2, a certain calculation and inversion are carried out to determine the pore structure characteristics in concrete.The principle of the low-field NMR instrument is to convert the energy change signal of water molecules at resonance into the T2 can be expressed by Equation (1) [The relaxation time tion (1) :(1)1T2=1TB2—the volume relaxation time of the fluid and S—pore surface area, V—pore volume, and TD2—diffusion relaxation time of fluid.In the formula: TB2 and TD2 can be ignored in saturated structures [In this experiment, saturated specimens are used, and ructures . AssuminG—pore shape factor, assuming that the pore structure is an ideal sphere, then G is 3 and r—is the aperture.In the formula: r can be determined from the T2 value measured by the low-field NMR instrument .When The specimen for this experiment has a cylindrical shape with a diameter and height of 50 mm. After reaching the specified curing age, the specimen is removed for vacuum saturation. The concrete specimen is first subjected to vacuum saturation for 8 h using a vacuum saturation device to ensure complete saturation. The specimens are then soaked in water for 24 h. The samples are then removed from the water saturation apparatus, wrapped in plastic wrap, and immediately placed in a pre-prepared water tank. Before the low-field NMR test is performed, the low-field NMR instrument is calibrated. The excess water on the surface of the plastic film is wiped off, and the samples are carefully positioned in the low-field NMR system for testing . The low2 and Al2O3 in RCP react with Ca(OH)2 to form cementitious products such as calcium silicate hydrate (C-S-H) and calcium aluminate hydrate (C-A-H). These products effectively fill the internal pores of the concrete, improving its compactness and promoting strength development [2, resulting in a reduction in the formation of hydration products such as C-S-H. Furthermore, the excess amount of RCP fills concrete microcracks without undergoing secondary hydration reactions, thereby weakening the mechanical properties of the concrete.The mechanical properties of RCP concrete and ARC at different curing ages are shown in elopment . Second,elopment , which delopment , resultielopment . Secondl2 is beneficial in stimulating the activity of RCP, thereby improving the mechanical properties of the concrete. The addition of Ca(OH)2 primarily increases the OH concentration in the slurry, which facilitates the formation of free unsaturated reactive bonds on the surface of RCP in an alkaline environment. As a result, the degree of polymerization of the SiO2 and Al2O3 network aggregates decreases, increasing their reactivity with the active components in the liquid phase of the system. Moreover, this process leads to an increased production of cementitious products, which ultimately stimulates the activity of the RCP and improves the mechanical properties of the concrete. Ca(OH)2, as an activator, introduces a significant amount of Ca2+ ions, replenishing the CaO content in the RCP and promoting the formation of silicate and other cementitious materials. Meanwhile, Ca(OH)2 can react with active SiO2, Al2O3 in a large amount of aluminum-silica glass in the RCP in a volcanic ash reaction to produce C-S-H and C-A-H with higher strength and water hardness.It can also be seen from 2 itself has certain cementing properties, but its solubility is low, resulting in insignificant early strength growth of the concrete. With the prolongation of the curing time, a large amount of Ca2+ and the rest of the active substances in the system were dissolved, and a large amount of calcium aluminate and some gelatinous substances were formed, which led to the improvement of the late strength of the system [The growth rates of ARC over RCP concrete mechanical properties at the same RCP loading and curing age are shown in e system .2S and C3S. The increased hydration results in the generation of a significant volume of C-S-H and C-A-H, which effectively fills the concrete pores and reduces the porosity of ARC [The variation patterns of RCP concrete and ARC porosity with RCP dosage at 28 d maintenance age are shown in y of ARC . Both RCy of ARC . On the y of ARC . In addi2, is not readily soluble in water, which hinders its ability to dissolve rapidly during the early stages of the hydration reaction [The variation pattern of porosity in NC, RC-1, and ARC-1 with curing age are shown in reaction . Over tiThe pore size distribution curves of concrete containing different amounts of RCP at a curing age of 28 days are shown in 2, the excitation of the secondary volcanic ash reaction of RCP leads to the formation of C-S-H gel. This reaction not only increases the strength of the concrete, but also effectively fills the internal pores, leading to an increase in the proportion of harmless pores and a decrease in the proportion of multiple harmful pores. As a result, the pore size distribution is significantly improved.According to the classification of pore classes in concrete by Wu , differeThe pore size distribution curves of NC, RC-1, and ARC-1 at different curing ages are shown in The pore size distribution trends of NC, RC-1, and ARC-1 at each curing age are shown in Determine the order of fundamental parameters:The relationship between the mechanical properties of RCP and fine-scale pore structure characteristics was investigated using gray correlation entropy analysis ,48,49. Tn = 7, m = 2, and l = 5.The mechanical properties of recycled powder concrete, denoted as 2.Dimensionless processing of data in each series:3.Calculate the absolute difference between the reference sequence and the comparison sequence.4.Determine the maximum and minimum values of the absolute difference, denoted as:5.Computing the number of correlation coefficients.6.Calculate the gray relation entropy.7.Calculate the gray entropy correlation grade.iY, and porosity, harmless pores, less harmful pores, multiple harmful pores, and harmful pores were selected as the comparison sequence jX. The parameter sequences are shown in The data of NC, RC-1, RC-2, RC-3, ARC-1, ARC-2, and ARC-3 specimens at the age of 28 d of maintenance were used as the basis for the theoretical calculation of the gray entropy correlation. Compressive strength and flexural tensile strength were selected as the reference sequence Based on From 8.Determine the system characteristic data sequence A and the sequence of related factors as:The gray prediction model is generated by accumulating the original series once so that the discrete data have a certain regularity, and the differential equation for the newly generated series is the GM prediction model, which is calculated using the least squares method to obtain the relevant parameters of the model . The GM 9.Find its corresponding once accumulated generating sequence (AGO).10.A is the immediate neighbor of B to generate a sequence of mean values.11.Establish the GM  model.Through the gray relational entropy analysis, the functional relationship between the system characteristic data sequence and the related factor sequence is established. The data of the top three pore characteristic parameters  and mechanical properties  of RCP concrete are used as the modeling set, and the data are substituted into the GM  gray prediction model. The compressive strength prediction model of RCP concrete is obtained by the Equations (11)–(15):Flexural tensile strength prediction model:After the GM  gray prediction model was established, the reasonableness of the model simulation and prediction performance was evaluated using accuracy tests such as the mean relative error and the posterior difference ratio . The criThe error test of the GM  prediction model of compressive strength and flexural tensile strength of RCP concrete is With the increase of RCP, the mechanical properties of concrete first increased and then decreased. When a reasonable amount of RCP (10%) is added, the secondary hydration reaction and micro-aggregate filling effect of RCP are beneficial to reduce the porosity of concrete and improve the mechanical properties of concrete. Excessive incorporation of RCP (30%) will cause the porosity of concrete to increase and the compactness to decrease, thereby reducing the mechanical properties of concrete.2 is beneficial to stimulate the activity of RCP, improve the pore size distribution of concrete, reduce the porosity, and improve the compactness, thus improving the mechanical properties of ARC. As the curing age increases, the RCP is fully hydrated, the pore size in the concrete is further refined, the structure becomes denser, and the mechanical properties are improved. At the same time, the effect of alkali excitation on the mechanical properties of concrete becomes higher and higher.The incorporation of 2.5% Ca(OH)The compressive strength and flexural tensile strength of RCP concrete are in the same order of gray entropy correlation degree of each pore characteristic parameter of low-field NMR, that is, harmless pore > porosity > multiple harmful pores > less harmful pore > harmful pore.The average relative errors of the GM  model for predicting the compressive strength and flexural strength of RCP concrete are 1.519% and 1.423%, respectively, and the accuracy is level 2. The posterior error ratios are 0.189 and 0.255, respectively, and the accuracy is one level. The established predictive GM  model can accurately predict the mechanical properties of RCP concrete.In this paper, the GM  prediction model is established by studying the compressive strength and flexural tensile strength of RCP concrete and analyzing the pore structure characteristics by low-field NMR. The main conclusions are as follows:


Ductility-based structural design is currently the mainstream method. In order to analyze the ductility performance of concrete columns with high-strength steel reinforcements under eccentric compression, corresponding experimental studies have been performed. Numerical models were established, and their reliability was verified. Based on the numerical models, the parameter analysis was carried out, where eccentricity, concrete strength, and reinforcement ratio were considered to systematically discuss the ductility of the concrete column section with high-strength steel reinforcement. The results show that the ductility of the section under eccentric compression increases with the strength of the concrete and eccentricity, and decreases with the reinforcement ratio. Finally, a simplified calculation formula capable of quantitatively evaluating the section ductility was proposed. In the 1950s, some scholars proposed the concept of ductility based on the nonlinear response analysis and research of structures, which summarized the seismic performance of structures or components beyond the elastic deformation stage ,3,4,5. TThere are many researches on the eccentric compression performance of columns, for examples, Chen and Chen et al.  experimeHigh-strength reinforcement can reduce the amount of steel used . There wResearchers have paid a lot of attention to the bearing capacity and seismic performance of concrete member with high-strength steel reinforcement, while few people holding on to the ductility of eccentrically compressed components. The ductility of members under load is insufficiently studied, and the ductility problem has been emphasized in the eccentric compression of concrete columns. The Eurocode EN 1998-3, 2005, contains expressions for the computation of the chord rotation capacity of RC members, and this paper will be a supplement to the quantitative research on the ductility of high-strength reinforced concrete columns under eccentric compression.In this paper, the ductility performance of six concrete columns with 500 MPa grade (the yield strength was 500 MPa) high strength steel reinforcement under eccentric loading was analyzed by the experimental test and the numerical models. The various influencing parameters including reinforcement ratio, eccentricity ratio and concrete strength on the ductility of the specimen were systematically studied, and a simplified calculation formula for evaluating the ductility of the concrete section with high strength steel reinforcement under eccentric compression is proposed.In the experiment, six concrete columns with 500 MPa grade high strength steel reinforcement under eccentric loading were manufactured. the length of columns are 1700 mm, and the cross-section of columns are all 300 mm × 400 mm, and the thickness of the protective layer is 25 mm for each column. The specimens were designed according to the Chinese code ; the detThe size of the test specimens and the configuration of the reinforcement are shown in A knife hinge was set up at the lower end of the column for simulating the boundary conditions of the hinged ends of the column under eccentric compression, and the actuating head at the upper end of the column was a ball hinge. A 40 mm steel bar was placed on the upper plate. The schematic diagram and the physical diagram of the loading are shown in The monotonic continuous loading strategy was adopted, and the load is from zero until the specimen fails. By estimating the yield load, the loading process was divided into two stages. The first stage was before reaching 80% of the yield load, which was controlled by the force. Each step of the first stage has a load of 50 kN and the loading rate was 100 kN/min. Subsequently, there was the second stage, and the displacement control was adopted, and each step was 1 mm of this stage. The load would be stopped when the load drops to 65% of the ultimate bearing capacity.The main measurement contents in the experiment include deflection, force and strain. The horizontal deflections were measured by LVDTs, while the vertical deflections were measured by the instrument itself. The strains of the reinforcements were measured by using a bonded strain gauge. The layout situations of the LVDT and reinforcement strain gauge are shown in The mechanical properties of the steel materials are shown in At the beginning of loading of each component, the load was small, no cracks appeared in the test specimens, the deflection and the strain of the steel bar and concrete increased linearly with the load, and the specimens showed elastic deflection characteristics. As the load increases, cracks began to appear on the tensile side of the specimens and began to widen and extend.For the specimens HRCC-1, HRCC-4, HRCC-5, and HRCC-6, when the load was close to the ultimate load, the tensile steel bar began to yield, and the cracks developed rapidly. The crack width of the tensioned surface reached 1 mm. Subsequently, vertical cracks appeared, and the concrete was crushed. The primary crushing area was at the mid-span section of the specimen, and the bearing capacity reached the maximum. Under continuation of the loading, the vertical cracks in the concrete on the side near the load quickly spread and penetrated, and the concrete protective layer was peeled off. When the load dropped to 65% of the ultimate load, the load was stopped, and each specimen has significant deflection.According to the failure form of the eccentric compression column, these specimens were damaged by large eccentric compression. The final failure pictures of the specimens are shown in For specimens HRCC-2 and HRCC-3, the crack width reached 1.5 mm, and the maximum length was about 25 cm when the load approached the ultimate capacity. The distance between cracks was about 8 cm. Subsequently, the concrete at the edge of the compression zone was suddenly crushed, and the central crushing zone was concentrated in the middle of the specimen. The load reached the ultimate bearing capacity and quickly dropped below 80% of the ultimate bearing capacity, and the loading was ended. The final failure pictures of the specimens are shown in The load-horizontal deflection curves at the midpoint of column of each specimen are shown in It can be found that under different reinforcement ratio parameters, HRCC-2 had a 21% increase in bearing capacity compared to HRCC-1, the deflection corresponding to the ultimate vertical bearing capacity increased by 23%, and the displacement decreased by 8% when the bearing capacity dropped to 85%.Compared with HRCC-5, the bearing capacity of HRCC-6 increased by 31%, the displacement corresponding to the ultimate bearing capacity increased by 16%, and the displacement decreased by 29% when the bearing capacity dropped to 85%. The reinforcement ratio had a greater effect on the bearing capacity, and the bearing capacity increases with the reinforcement ratio. After the specimen cracked, the deflection of the specimen with a large reinforcement ratio increased rapidly, and the specimens with a small reinforcement ratio under the same load had a larger deflection. After the HRCC-1 and HRCC-5 reached the ultimate bearing capacity, the applied load decreased more gently, and the ductility was better than that of the specimen with a large reinforcement ratio.Under different concrete strength conditions, the bearing capacity of HRCC-1 was 0.4% different from HRCC-3, and HRCC-6 was 7% higher than HRCC-4. It can be seen that the concrete strength within the test parameters had no effect on the bearing capacity, while after the specimens cracking, the specimens with higher concrete strength had greater stiffness than that with lower concrete strength.Under different eccentricity conditions, the bearing capacity of HRCC-5 was 22% lower than that of HRCC-1, and HRCC-6 was 16% lower than HRCC-2. It can be seen that within the range of the experiment parameters, the eccentricity ratio had a great influence on the specimens, and the bearing capacity decreased with the increase of the eccentricity ratio. This is because the specimens with a large eccentricity had a smaller depth of compression zone, which reduces the vertical load-bearing capacity. After cracking, the specimens with large eccentricity had less rigidity and larger lateral deflection under the same load. This was because the concrete with large eccentricity cracks faster in the compression zone, which leads to faster deflection.Whether the tensile steel reinforcement yield first during failure can be regarded as the judgment basis for eccentric compression. Due to the limited experimental research, numerical models were created to make systematic evaluations for concrete columns with high strength steel reinforcement under eccentric loading. The sectional numerical models were established by using XTRACT  softwaresR of concrete cubes used in China’s concrete code as cf′ = 0.85 sR. The strength of the concrete material corresponding to the model in the test are given in The stress-strain relationship of concrete adopts the default unconstrained Mander model. The 28-day standard value of compressive strength of concrete cylinders is related to the standard value of compressive strength Based on the above material stress-strain relationship, the same numerical analysis model as the test parameters is used to calculate the loading process. The moment and curvature of the specimen can not be directly obtained during loading, where the moment was obtained by the load and eccentricity, and the curvature was converted from the data collected by the strain gauge pasted on the reinforcement, and it can be written as,The moment-curvature curve obtained from the numerical calculation results is compared with the test results, as shown in Since the curvatures were obtained through strain conversion, and strain testing is not stable, the experimental moment curvature curves of some specimens, such as HRCC-2 and HRCC-5, are not ideal. HRCC-3 experienced instrument failure during the loading process of the test, the curve after the bearing capacity decreased failed to captured. However, the test curve before reaching the ultimate bearing capacity was close to the numerical curve, and the numerical results could compensate for the shortcomings of the test. However, overall, the numerical results curves were consistent with the experimental results curves.yφ denotes the yield curvature, for components with obvious yield points, this value is relatively easy to determine, but for components without obvious yield points, some measures need to be taken, for examples, Colajanni et al. [In order to more accurately analyze the eccentric pressure performance of HRCC columns, XTRACT was used in a parameter analysis. 102 kinds of section of concrete columns with high strength steel reinforcement under eccentric loading with different concrete strengths, reinforcement ratio, and eccentricities were designed. The concrete strength grades were C25, C30, C35, C40, C45, C50, C55, C60, C65, C70 total 10 cases; There were 9 types of reinforcement ratios: 0.34%, 0.42%, 0.52%, 0.63%, 0.71%, 0.82%, 0.95%, 1.05% and 1.23%; The eccentricity had a total of eight cases of 0.475, 0.525, 0.575, 0.625, 0.6975, 0.75, 0.8 and 0.85. The dimensions of each section were the same as those in the test and are all 300 mm × 400 mm. The ductility factor was calculated the following formula,i et al.  and Palai et al.  proposedBased on the analytical results of concrete parameters, the influence of different concrete strengths on the ductility capacity of concrete columns with high strength steel reinforcement under eccentric loading with different eccentricity and reinforcement ratio is shown in Overall, the strength of concrete had increased from 20 MPa to 45 MPa, the ductility of cases exhibits varying degrees of increase, in which the maximum growth rate reached 61%. It can be concluded that in the design of section of concrete columns with high strength steel reinforcement under eccentric loading, adjusting the strength of the concrete relative to adjusting the reinforcement ratio has no significant effect on the ductility of the section. In the case of proper reinforcement ratio and eccentricity, the ductility of sections can be improved by properly increasing the concrete strength.ξ is the relative depth of compression zone, and it can be calculated by the following formula,x denotes the depth of compression zone, and when the section is subjected to large eccentric compression failure, it can be calculated by the following formula,e is the eccentricity.According to the previous analysis, it can be concluded that ductility capacity of concrete columns with high strength steel reinforcement under eccentric loading is related to the longitudinal reinforcement ratio, concrete strength, and eccentricity ratio. The sensitivity of ductility to each parameter is in the order of reinforcement ratio, eccentricity ratio and concrete. The section ductility decreases with the increase of reinforcement ratio, increases with the increase of eccentricity, and increases with the increase of concrete strength. Regression analysis is performed based on the results of the section ductility coefficient obtained from the numerical analysis, and the mathematical expression is as follows,x can be calculated by the following formula,When the section is subjected to small eccentric compression failure, 010-2010 .The calculation result of the regression formula is shown in (1)The failure modes of concrete columns with high strength steel reinforcement under eccentric loading are similar to that with ordinary steel reinforcement.(2)The ductility of the column section under eccentric compression increases with the increase of concrete strength, but in some cases there was no obvious pattern.(3)The ductility of column section decreases with the increase of reinforcement ratio, especially when the eccentricity is large, and the ductility of the sections with larger eccentricity is generally larger. For small eccentricity, when the reinforcement ratio increases from 0.34% to 1.23%, the ductility decreases by about 58%, and for large eccentricity, the ductility decreases by about 70%.(4)The ductility of column section increases with the increase of eccentricity ratio, and the change is greater when the reinforcement ratio is smaller. For a smaller reinforcement ratio, when the eccentricity increases from 0.475 to 0.85, the section ductility increases by about 113%, and for a larger reinforcement ratio, the ductility increases by about 45%.(5)The simplified calculation formula of curvature ductility can accurately evaluate the ductility of the concrete columns with high strength steel reinforcement under eccentric loading.(6)The ductility of the concrete columns with high strength steel reinforcement under eccentric loading can be improved by improving the concrete strength and increasing eccentricity ratio. Increasing the reinforcement ratio has a negative impact on ductility, so when there is a demand for high ductility, it is necessary to control the reinforcement ratio to not exceed a certain limit value.(7)There are some limitations in the work of this manuscript, for examples, the results were based on cross-sections and ignored the influence of slenderness ratio, and the bidirectional eccentricity was not considered. In future research, more attention will be paid to spatial effects.The ductility capacity of concrete columns with high strength steel reinforcement under eccentric loading was investigated. The ductility was analyzed based on experiments, and the numerical model was estimated, and the parameter analysis was performed. Through extensive analysis of the results, the influence of various parameters on ductility was obtained, and a calculation method for ductility coefficient was proposed. Some conclusions can be drawn as follows,


The Drift-Diffusion Model (DDM) is widely accepted for two-alternative forced-choice decision paradigms thanks to its simple formalism and close fit to behavioral and neurophysiological data. However, this formalism presents strong limitations in capturing inter-trial dynamics at the single-trial level and endogenous influences. We propose a novel model, the non-linear Drift-Diffusion Model (nl-DDM), that addresses these issues by allowing the existence of several trajectories to the decision boundary. We show that the non-linear model performs better than the drift-diffusion model for an equivalent complexity. To give better intuition on the meaning of nl-DDM parameters, we compare the DDM and the nl-DDM through correlation analysis. This paper provides evidence of the functioning of our model as an extension of the DDM. Moreover, we show that the nl-DDM captures time effects better than the DDM. Our model paves the way toward more accurately analyzing across-trial variability for perceptual decisions and accounts for peri-stimulus influences. When decisions are timed, evidence accumulation models accurately describe human and animal behavior. They assume that decisions are made when enough sensory evidence has been gathered.Perceptual decision-making has been studied extensively from behavioral5 suggests that evidence is accumulated linearly, with a constant drift. The accumulation is additionally subject to Gaussian noise; hence the decision state can be seen as a particle following a Brownian motion. The popularity of this model yields from its intuitive formalism and good fit to behavioral1 and neurophysiological data3. It has also been shown that the DDM formalizes the optimal strategy for decision-making under time constraints7. Interestingly, other forms of decision models such as the Leaky-Competing Accumulator model8, and attractor models10 can be formulated equivalently to the DDM under certain constraints11.Among them, the Diffusion Decision Model 12 impact prior expectations and the subsequent decision process13. Traditionally, expectations are modeled through the starting point, or bias, of the accumulation process5. Recent accounts have suggested that choice history affects subsequent drift rates14. Together, these studies suggest that these parameters could be intertwined and vary over time, as participants become more familiar with the task. To address this issue,16 proposed an extended DDM, where starting points are uniformly distributed and drifts follow a Gaussian distribution without explicit dependence between them. However, this only provides global statistics about perceptual responses without insight into single decisions or inter-trial interactions. Moreover, the linear accumulation does not describe the variation of the dynamics at the scale of the single decision, which seems inconsistent with the aforementioned empirical observations.The DDM accounts for global statistics of the behavior by describing the Response Time (RT) distribution and error rate. A major limitation of this model is that it does not consider inter-trial variability. However, behavioral studies have shown that sequential effects17, or dynamics close to the threshold19, no model to our knowledge accounts for initial dynamics. For example, ambiguous stimuli could yield flat initial drifts. This is partially translated into non-decision time, as it is assumed to be when sensory evidence is processed in the brain without contributing to the decision process.Linear evidence accumulation also assumes that evidence accumulation is independent of the decision state or of the time that passes. While some models consider the effect of time on the decision21, and it has been shown that these models can be reduced to a Drift-Diffusion Model22, that is in that case, a Langevin equation with a non-linear drift23. Its dynamics allow for transitions between decision states under fluctuating stimuli11. However, the link between each parameter and the dynamics of the model is complicated to interpret. Moreover, the reduction proposed assumes a reflection symmetry of the network to obtain the given form. This seems limiting, in particular when each perceptual decision recruits different sensory modalities.Previous attempts at single-trial fitting of decisions have been made through attractor models24. Increased environmental requirements in terms of workload can decrease RTs and alter accuracy25. While some models have taken into account the passing of time within each trials17, no models have tried to account for more global fluctuations to our knowledge.Moreover, while the few parameters of the DDM are advantageous in terms of complexity, it can be a limiting factor when analyzing endogenous effects, such as fatigue or training on decisions. Previous works have shown that post-stroke fatigue increases the non-decision time along experiment time, while RTs tended to decrease in healthy participants26, and a multisensory classification task. Then, we used correlations to compare the parameters of both models on data simulated from DDM parameters and provide an empirical explanation of the effect of the nl-DDM parameters with analogies on the DDM. We show that it fits data equally well as the DDM while providing drift variability like the extended DDM. The dependency of the drift rate on the decision state provides a framework for more refined analyses of the decision process. Last, we considered the time spent performing the lexical task and showed that the nl-DDM modeled behavioral data significantly better than the DDM in that instance, supporting the necessity to account for the experiment time. We provide open-source code pluggable onto the PyDDM toolbox22 for reproducibility and easy use of our model.Here we propose a straightforward one-dimensional non-linear form to address these limitations: the non-linear Drift-Diffusion Model (nl-DDM). It recreates double-well-like dynamics from an evidence-accumulation perspective without assuming reflection symmetry. We show its validity and compare its fitting performances to these of the DDM. We first provide a formal description of the nl-DDM, relating it to the DDM. Then, we fit the models on two human behavior datasets: a lexical classification task already published26, on which we also modeled the effects of the time spent doing the experiment, and a multi-sensory classification task. To provide insight into the meaning of nl-DDM parameters, we also performed correlation analyses between nl-DDM and DDM parameters on data generated from DDM parameters and subsequently fitted by the nl-DDM.In this paper, we introduce the non-linear Drift-Diffusion Model (nl-DDM) and show that it performs better than the DDM in terms of fitting accuracy on behavior. To this aim, we fitted both models on two datasets: a lexical classification dataset previously published in27.Our goal was to propose a simple model in which trajectories are attracted to a boundary. Placing ourselves in the context of two-alternative choice paradigms, our model needed two attractive states. In one dimension, this forces the existence of an unstable fixed-point between the two stable fixed-pointsx represents the decision variable and dx its variation in infinitesimal time dt. N(t) is a Gaussian white noise term. k is a time constant of the system, and a and z determine where the attractors, or decision boundaries, lie. z to the interval z the unstable fixed-point. In this case, the drift corresponds to the deterministic part of the equation and depends on the current decision state. A summary of the parameters of the nl-DDM is given Fig. Therefore, the model we propose follows a Langevin equation where the drift varies with the state of the decision. The drift equation can be written in the following form:k as a time constant is straightforward from the equation: as k increases, a decision is reached faster for any given set of parameters.The interpretation of To build an intuition for the other parameters, we first consider the potential function derived from the drift term Fig. :2\documeThis profile is called a double-well potential profile.z, which emerge from the topology of the system. Therefore, z, the speed-accuracy trade-off. Taking a the boundary for correct responses and z closer to a-well deeper , which emerges from attractor network models27. However, when Comparing this equation to Eq. , we obsez and This equivalence is coherent with the interpretation of It is helpful to obtain each participant’s RTs and decision accuracy for decision-making analysis, particularly for decision model fitting.26, in which participants had to assess whether a word presented on screen existed or not. The second one is a dataset in which participants were shown visual stimuli or a combination of visual and auditory stimuli on screen and had to classify them according to their type (either “face” or “number+sound”).We used two datasets in this paper, described in the Methods section. They both consist of classification tasks performed by human participants. One of them is a dataset collected by Wagenmakers et al.26 are discussed later.We describe here the validation conducted on the multi-sensory dataset. Analyses of the lexical datasetOn average, participants were shown mentclass2pt{minimIn the “face-left” stimulus-response mapping, where participants were instructed to click left upon face stimulus presentation and right when they were presented with a number+sound stimulus, participants responded on average within mentclass2pt{minimThese results show the uniformity of participant responses across mappings and stimuli.22 for both the nl-DDM and the DDM, minimizing the negative log-likelihood function. We fitted a model per participant and model type, resulting in 25 DDM and 25 nl-DDM on the sensory classification dataset. The DDM was fitted using 6 parameters , and the nl-DDM consisted of 7 parameters , 1 starting point and variability, 1 non-decision time).Parameter fitting was performed using PyDDMt-test, We computed the Bayesian Information Criterion (BIC) for each model fitted on the multi-sensory dataset to establish a comparison of model performance that considers the sample size and number of parameters for each model. This is indeed necessary when comparing performance across model types, since the number of parameters is different. We observed that the nl-DDM fitted RT data significantly better than the DDM, even when accounting for the number of parameters  or the second (“late” condition). Participant 2 did not complete all the blocks, and was eliminated from our analyses. Therefore, the following analyses are presented over 16 subjects, each exposed to all the conditions.z of the DDM and nl-DDM varied as a function of the stimulus complexity , and the boundary of the DDM varied depending on both the instruction (speed/accuracy) and the time condition, resulting in 4 boundaries. In the nl-DDM the effects were modeled separately using a and k, fitting k depending on the time condition and a according to the instruction. Then, the starting-point distribution and non-decision time were fitted over all trials for each model. We compared the fitting performance of the two models. The BIC of the nl-DDM was significantly smaller than the BIC obtained by the DDM (Shapiro-Wilk test: The drift and z parameters (one per word type), 2a (one per instruction) and 2k (one per time of the experiment). We then performed paired t-tests to assess the discriminability of the parameters across conditions. We first compared the drift and z parameters across word types. In both the DDM and nl-DDM, we observed significant differences in the drifts and z between all word type pairs, except between rare and non-existent words  and the word type for each trial. In the DDM, we hence fitted 4 drifts, corresponding to the 4 word types, and 4 boundaries, corresponding to s Tables , 6. The We presented a non-linear model of decision-making. This model takes the form of a Langevin equation, and provides a framework in which individual trajectories of the decision variable can have different shapes under the same global parameters Fig. D. We havz towards the negative boundary in the nl-DDM, as they both result in fast correct responses. However, note that these parameters are not entirely equivalent as we did not find a perfect mapping between them, meaning that the nl-DDM is conceptually different from the DDM . The version we showed here only split the trials into two instruction and time conditions, which results in the same number of parameters in the DDM and in the nl-DDM. Note however that any more instances of either condition would have meant more parameters in the DDM relative to the nl-DDM.We have shown that the nl-DDM can also account for changes in the decision dynamics that do not relate directly to a change in the experiment but rather to the state of the participants. While such an account necessitates an exponential multiplication of parameters in the DDM, the nl-DDM requires a simple duplication of the parameters. Indeed, if we again consider the case where there are two conditions , the DDM requires 32. EEG research has shown a matching between pre-stimulus activity and confidence ratings in human participants34. The starting-point distribution models pre-stimulus states, and in the DDM the drift relates to the quality of the integrated stimulus4, with more ambiguous stimuli corresponding to lower drift rates. At the single-trial level, drift variability relates to the variation of the quality of stimulus perception and processing in the brain1. In our model, the starting point directly impacts evidence accumulation, allowing for a uniform theory of decision-making that includes explicit co-dependency of parameters. Some general forms of the DDM include a variance of the drift. In the nl-DDM, we have not implemented this possibility, as we assumed that the inter-trial variability of the drift emerged from the variability of the starting point. In neurophysiological terms, we assumed that the pre-stimulus arousal and stimulus expectations led to differences in the rate of evidence accumulation. This is supported by past observations, according to which pre-stimulus brain activation impacts RTs36. Pre-stimulus brain activity also modifies perceptual37 and pain38 thresholds. Therefore, depending on the pre-stimulus activity, decisions can be made even in the absence of actual evidence39, or under ambiguous evidence41. Along the same lines,42 have shown that biases were implemented through local changes in accumulation rate, which supports the intertwining of accumulation rate and pre-stimulus states. However,45 argue that pre-stimulus brain states should only affect the decision criterion, not how well participants perceived the stimuli. Translating the signal-detection theory to the evidence-accumulation scheme16, pre-stimulus states should only be changing the decision boundary, or equivalently the starting point, and not the drift rate. For example,34 found that pre-stimulus alpha power did not impact the accuracy of visual evidence accumulation, but only the confidence in the decision.33 found similar results with auditory stimuli. Although these observations seem to contradict our assumption that the starting point impacts the evidence-accumulation process, both phenomena could co-exist, as more extreme starting points are more attracted to the closer attractor. This results in fast and confident observations, although little evidence has been accumulated (we would be located at a plateau in our model), i.e., even if the stimulus was not well perceived.We argue that drift and starting-point variability are intertwined, as transcribed in the nl-DDM and in alignment with the view that evidence accumulation starts in anticipation of stimulus apparition29. The reason is two-fold. First, we wanted to use simple forms of both models to emphasize the characteristics of the ground parameters of each model. One may argue that the DDM should then have been fitted using a single point as the starting point. However, this would have introduced a confound when comparing the two models. Indeed, a major advantage of the nl-DDM is the variety of dynamics that it offers depending on the starting point. Since the DDM can also be improved by adding starting-point variability, implementing it in both models seemed to be a fair compromise. Second, and related to the first, any source of variability that could have been introduced in the DDM could also have been implemented in the nl-DDM. Besides starting-point variability, non-decision time variability could also exist in the nl-DDM. As argued above, the drift and starting-point variability should be intertwined, meaning that fitting a variability of the drift would be redundant to some extent in the nl-DDM, but we could implement variability in k, which, according to our analyses on time that passes, could be a natural mirror of the remaining effects of drift variability in the DDM.The current analysis uses a form of the DDM without all variabilities proposed by Ratcliff and Tuerlinckx35 and36 have shown in human EEG studies that the brain activity prior to stimulus presentation changed the speed of responses. More specifically, they showed that the more pre-activated the required sensory area, the faster the decision. The nl-DDM mimics this behavior at the single-trial level: for trials starting close to the unstable fixed-point , the trajectories start with a plateau-like stage, whereby little evidence is accumulated because the brain would need to process the stimulus more intensively in order to extract information from it, before integrating evidence faster. This initial inertia is circumvented by shifting the starting point closer to the decision well, resulting in faster and more accurate responses. The initial inertia in the DDM is referred to as the non-decision time and encompasses both sensory processing and motor planning and execution. The nl-DDM assumes therefore that part of these processes participates in the decision process, which goes beyond the conceptualization of decision-making as the sequence of sensation, perception, and motion.The dynamics that we propose here is rooted in empirical observations made in neurophysiological studies. More specifically, three phases can be identified in the decision trajectories: an inertia stage, a quasi-linear evidence accumulation stage, and a plateau stage. The initial inertia relates directly to the brain activation needed to integrate sensory evidence.46 shows the limitations of existing evidence-accumulation models. We try to address several of them with the nl-DDM, including the possibility for analyses beyond the global description of RTs and the formulation of initial and final dynamic changes during the decision process. In particular, our formal description has shown that different shapes of decision trajectories can co-exist within the same framework, not solely because of noise, but because of meaningful variability. We expect this model to be further used to gain insight into the across-trial variability of decisions.A recent review from48. In non-stationary contexts, the input can be considered as a variation of z in time. By shifting z to either boundary, more trajectories are attracted to the opposite boundary, hence increasing the likelihood of correct answers. In addition, it can be inferred from our formal analysis that changing z means changing the drift rate. This change in input could also explain error-correcting behaviors49 and spontaneous changes of mind50. When the stimulus ends, the DDM is modified so that the drift is null, i.e. evidence is no longer accumulated. Therefore, changes of mind are the result of noise in the system. Conversely, stimulus termination could be modeled through shifting z in the nl-DDM, which effectively modifies the drift rate of the current decision, in a way that the decision variable could toggle towards the opposite boundary upon stimulus disappearance. Conceptually, the drift in the nl-DDM not only relates to the accumulation of evidence but also encompasses decision processes related to the post-processing of evidence.The current study considered that the input was presented at the beginning of the trial and affected the decision in a constant fashion. We could also imagine more dynamic cases, where the input is processed over a finite period and participants accumulate evidence during stimulus presentation, as has been done in past DDM analyses11 derived from attractor models23, the nl-DDM is equivalent to it only in the absence of input. A question that remains open is that of the mechanism underlying this equation. From the reduction computed in the paper by23, it would seem that a network of three populations could produce the dynamics we have described. However, the main assumption of the reduction was that the network was invariant through reflection. We argue that the mechanisms described by the nl-DDM are similar to these of the DWM, but offer a broader range of applications beyond the case of symmetrical models.We have shown that, while similar to the DWMD-space, adding more stable fixed-points will result in two issues. First, traveling from one alternative to another may require passing through other decision wells, which seems incoherent with behavior. Second, adding stable fixed-points requires the implementation of as many unstable fixed-points, which would mean a two-fold increase in the number of parameters when adding one choice. A simpler solution would be to switch to a 2D-space, so there could be a central unstable fixed-point, and the subjective preference for each alternative would determine the position of each stable fixed-point.Extending this model to multiple-choice situations is another interesting ground of research. The DDM is inapplicable in such situations. The nl-DDM would require structural changes for multiple choices. Indeed, the decision variable’s trajectory is here modeled in a one-dimensional space, where the alternatives are represented as attractors. Its multiple-choice variant would require more attractors. In 15 is characterized by a linear accumulation disturbed by additive noise. Formally, this can be written as the following Langevin equation  is a Gaussian white noise, parameterized by its standard deviation The Drift-diffusion modelion (Eq. ):4\documEvidence is accumulated following Eq.  until a 51, or from other parametric distributions15. We will consider uniformly distributed starting points in our fitting to provide a fair comparison of the two models without loss of generality.Finally, the starting point of accumulation is called the bias and is defined as a single point within the two boundaries. In general forms of this model, it is also possible to consider that the starting point is drawn from a uniform distribution centered around the bias The boundary separation represents the speed-accuracy trade-off. Indeed, if this separation is bigger, decisions are less impacted by noise and hence more accurate, but at the same time, they will take longer to reach from a given starting point. In contrast, the drift mainly impacts the speed of response, as a higher drift will lead to faster correct responses and longer incorrect responses.Fitting is typically done globally over RTs. In fact, the trajectories defined by the equation cross the decision boundaries, forming a RT distribution usually compared to an exponentially modified Gaussian. In order to obtain a close fit, it is necessary to define a non-decision time  healthy right-handed participants aged Participants were presented at each trial with images of faces or images of numbers, and had to respond with mouse clicks to report what stimulus they perceived. A sound accompanied images of numbers to suppress any ambiguity. Participants were instructed to respond using their right hand. To control for possible differences in motor response speeds between the two fingers, one group of participants . In total, stimuli were split into 4 categories of interest. Each participant performed 20 blocks of 96 trials each, with as many invented words as real words in each block. Participants were given the additional instruction to define the speed-accuracy trade-off in each block: they alternated between blocks where speed was emphasized and blocks where accuracy was more important. Responses were limited to 3 s, and trials with RTs below 180 ms were discarded to avoid anticipatory responses. More details can be found in26, and the dataset can be accessed from here.To model the effects of time passing and discard the possibility of better performances emerging from the fitting algorithm or data acquisition, we also lead our analyses on a bigger pre-existing dataset taken fromWe are interested in comparing model parameters between the DDM and the nl-DDM. It is important to check whether participants’ performance across stimulus-response mappings and stimuli is coherent in terms of RTs and accuracy. Indeed, the multi-sensory experimental paradigm we defined entails two types of stimuli and two motor commands for the choices. In addition, we have created two experimental groups, which were instructed to respond with opposite motor commands. First, we computed the percentage of stimuli in each class to verify that the stimuli were globally equiprobable for each participant. Since we designed the experiment to display each stimulus with the same probability at each trial, we expect this number to be close to 26, the effect of the time of the experiment is of special importance, as well as its interaction with the other experimental conditions (i.e. the word type and the instruction). We therefore performed two repeated-measures ANOVAs, respectively testing RTs and mean accuracy, assessing the effects of time , stimulus frequency, and instruction (accuracy or speed). One of the participants (participant 2) did not perform the In the lexical classification datak, the middle of the starting-point distribution 5, since the speed-accuracy trade-off is determined by the boundary separation, fitting two parameters among drift, boundary, and noise is constraining enough. In addition, each model requires fitting a non-decision time The classical way of fitting evidence-accumulation models is by fitting one drift for each stimulus category separately. In that case, the positive and negative boundaries still correspond to correct and incorrect responses respectively, and the starting points are taken from the same distribution regardless of the stimulus. Consequently, one pair of boundaries 22 for the fitting, minimizing the negative log-likelihood function and an implicit resolution. The nl-DDM indeed does not have explicit solutions when z is not centered. The log-likelihood is such that the more negative, the closer the modeled distribution of RTs is to the empirical RT histogram.We used the PyDDM toolboxWith this dataset, we were interested in modeling the effects of time passing throughout the experiment. Therefore, we created an artificial condition based on the sequence of blocks, which characterized the trials as happening early  or late . We discarded participant 2, for whom we did not have data for the z per stimulus type (a was fitted per instruction condition and one parameter k per time condition, resulting in 4 more parameters. Similar to the DDM, the non-decision time, starting point, and starting-point variability were fitted over all trials from each participant and the noise scale is set to 0.3.Within the nl-DDM framework, we fitted for each participant one 22 with negative log-likelihood minimization and implicit resolution.As previously, we used PyDDMSince the fitting on both datasets was performed using a different number of parameters and samples, we computed the Bayesian Information Criterion for each model, defined as:t-test. Indeed, we want to test whether the nl-DDM is better than the DDM with these three metrics, hence testing the hypothesis Hence, we compare each loss pairwise, using a one-sided paired-sample For a better understanding of the parameters of the nl-DDM, their interaction and their meaning in the DDM framework, we computed the Pearson’s correlation coefficients of DDM and nl-DDM parameters over simulated experiments. In order to obtain correlation coefficients within nl-DDM parameters as well as across DDM and nl-DDM parameters, we varied one by one the DDM parameters We computed the correlation matrix between all the parameters of both models. This allows for a first look into first-order interactions between model parameters, within and across model types. Since the correlations within DDM parameters were irrelevant due to their artificial manipulation, these coefficients were not computed. The correlation coefficients were computed using Pearson’s Supplementary Information.


We conclude that beta-lactam antibiotics can be detected and measured from the exhaled breath condensate accumulated in the HMEF from mechanically ventilated patients. However, no correlations were observed between the antibiotic concentrations in HMEF with either plasma or ELF.Different techniques have been proposed to measure antibiotic levels within the lung parenchyma; however, their use is limited because they are invasive and associated with adverse effects. We explore whether beta-lactam antibiotics could be measured in exhaled breath condensate collected from heat and moisture exchange filters (HMEFs) and correlated with the concentration of antibiotics measured from bronchoalveolar lavage (BAL). We designed an observational study in patients undergoing mechanical ventilation, which required a BAL to confirm or discard the diagnosis of pneumonia. We measured and correlated the concentration of beta-lactam antibiotics in plasma, epithelial lining fluid (ELF), and exhaled breath condensate collected from HMEFs. We studied 12 patients, and we detected the presence of antibiotics in plasma, ELF, and HMEFs from every patient studied. The concentrations of antibiotics were very heterogeneous over the population studied. The mean antibiotic concentration was 293.5 (715) ng/mL in plasma, 12.3 (31) ng/mL in ELF, and 0.5 (0.9) ng/mL in HMEF. We found no significant correlation between the concentration of antibiotics in plasma and ELF (R Pneumonia is a frequently encountered complication in critically ill patients and is known to be linked with significant morbidity and mortality rates . The timWhile measuring antibiotic plasma levels is a common practice in the ICU, it may not accurately reflect the actual concentration of antibiotics in the lung parenchyma. Following the intravenous administration of a beta-lactam antibiotic, the plasma concentration undergoes a characteristic rise and subsequent exponential decline. However, it is important to note that the peripheral compartments, such as the cellular and interstitial compartments, may exhibit different rates of change in their drug concentrations. The equilibrium between these compartments does not occur instantaneously, leading to potential differences in the concentration-time curves of the compartments. Despite these differences, plasma levels remain the primary clinical tool for guiding clinicians in adjusting antibiotic dosages.Various techniques have been proposed for assessing tissue concentrations, including pulmonary microdialysis , tissue An alternative non-invasive procedure for assessing lung conditions is the evaluation of exhaled breath condensate (EBC). This method has proven to be useful in various scenarios, including the categorization of acute respiratory distress syndrome (ARDS) severity  and the Therefore, the objective of the present study is to investigate whether the concentration of antibiotics measured in exhaled breath condensate collected from HMEF correlates with the concentration of antibiotics measured in the ELF from bronchoalveolar lavage (BAL) samples. This observational clinical study is designed to include mechanically ventilated immunosuppressed patients suspected of having pneumonia, for whom a BAL procedure is conducted to confirm or rule out the diagnosis of pneumonia. These patients are concurrently receiving beta-lactam antibiotics as part of their treatment regimen.We conducted an observational clinical study in the Critical Care Unit of Hospital Clínico UC-Christus  between January 2019 and January 2020. The study was designed in accordance with the guidelines of the Declaration of Helsinki, and it received approval from the Research Ethics Committee of the School of Medicine at Pontificia Universidad Católica de Chile . Written informed consent was obtained from all patients to publish the findings of this study.The study included consecutive patients who were connected to invasive mechanical ventilation and required a fiberoptic bronchoalveolar lavage (BAL) as indicated by the attending physician. Furthermore, these patients received beta-lactam antibiotics for a minimum duration of 6 h . ExclusiOnce the patient was connected to mechanical ventilation, an HMEF  was installed into the ventilator circuit, specifically positioned between the Y-piece and the endotracheal tube. The HMEF utilized in our study possessed an internal volume of 49 mL and was composed of a cellulose-based heat and moisture exchanger (HME) in combination with an electrostatic polypropylene filter.To ensure accurate and representative samples for analysis, we kept the HMEF in place for a minimum duration of 6 h before removing it from the ventilator circuit. This time interval was designed to allow sufficient time for the HMEF to capture and retain enough exhaled breath condensate and antibiotics within its structure for subsequent analysis.The selection of the appropriate beta-lactam antibiotic was at the discretion of the attending physician. The antibiotics were administered via an intravenous infusion according to previously published guidelines ,18.We included patients receiving ceftriaxone , piperacillin-tazobactam , imipenem , or meropenem . According to the ICU protocol, the beta-lactam antibiotics were administered starting with a loading dose followed by prolonged infusions over 3–4 h, except for ceftriaxone which was administered as a daily bolus.The bronchoscopic procedure was performed by an expert bronchoscopist. The tip of the bronchoscope was positioned in a subsegmental location, specifically targeting the area suspected of having a pneumonic process based on radiological findings. Bronchoalveolar lavage (BAL) was then carried out by sequentially instilling and recovering five separate 30 mL aliquots of 0.9% NaCl solution.One of the BAL fluid (BALF) aliquots was promptly separated and transported to the laboratory. Upon arrival, the sample was centrifuged, allowing for the collection of the supernatant. This supernatant was then stored at a temperature of −80 °C until further analysis.Considering that the introduction of the saline solution during the BAL procedure may result in a dilution effect on the BALF, we employed the urea concentration in both plasma and BALF to correct for this effect . By utilg revolutions per minute (rpm) for 5 min, and then the supernatant was collected and stored at −80 °C until analysis.Simultaneously with the bronchoalveolar lavage (BAL), plasma samples were obtained from the patients. The samples were transported to the laboratory and centrifuged at 3500× During the preparation for BAL, HMEF was removed from the ventilatory circuit; both inspiratory and expiratory ports were carefully sealed to prevent the evaporation of the condensate and immediately transported to the laboratory using a refrigerated unit. To extract the condensate from the HMEF, the tube connection port of the HMEF was introduced into a sterile 50 mL falcon tube and securely fixed in place while ensuring the distal port remained sealed. Subsequently, the falcon tube plus the HMEF were centrifuged at 4500 rpm for 15 min at a temperature of 4 °C , allowing the separation of the condensate from the HMEF. The supernatant was collected and stored at −80 °C until analysis. In order to have a negative control for comparison, HMEFs soaked in 0.9% NaCl solution were used. These HMEFs served as a baseline to differentiate any potential contaminant present in the HMEF that could have been mistaken as beta-lactam antibiotics.To determine the concentrations of beta-lactam antibiotics in our study, we employed a fluorescent biosensor known as Pen-Pcf20. This biosensor is structurally related to beta-lactamase but lacks catalytic activity. When Pen-Pcf20 interacts with the antibiotic present in the medium, its intrinsic fluorescence undergoes changes over time, generating a characteristic pattern that allows for the identification of the specific antibiotic being analyzed. For each antibiotic, the fluorescence curves obtained by increasing antibiotic concentrations were analyzed. These curves were fitted to a 4-parameter sigmoid curve (4PL) with a variable dynamic range, which depends on the particular antibiotic being tested. Generally, the dynamic range spans from log −9 to log −3, with an inflection point typically around log −6. In cases where the fluorescence of a sample was very high or saturated (>log −6), the sample was successively diluted by a factor of 10 (titrated) until its fluorescence fell within the dynamic range of the technique. This dilution process allowed us to accurately determine the antibiotic concentration via extrapolation . Samplesp < 0.05 was considered statistically significant. We analyzed the correlation among the continuous variables using the Pearson correlation coefficient. The analysis was performed with GraphPad Prism version 8.00 for Mac .Since there were no previous data regarding antibiotic levels in HMEF, and as this was an exploratory study, we did not perform a sample size calculation, and we arbitrarily decided to include 12 patients. The Shapiro–Wilk test was used to test data for normality. We expressed values as mean-standard deviation (SD) or median-range (IQR), where appropriate. We compared antibiotic levels between the three compartments analyzed  using a one-way (repeated measures) analysis of variance (ANOVA). n = 7). The mean age of the patients was 44 (20.9) years, and the mean body mass index (BMI) was 25.2 (6.9). All patients presented with respiratory failure upon admission to the intensive care unit and had immunosuppression as their primary diagnosis. In our analysis, we included a total of twelve patients, with the majority being female participants  μL of exhaled breath condensate was collected from the HMEF. Antibiotics were detected in the plasma, exhaled breath condensate collected from HMEF, and BALF samples of every patient studied. In addition, we observed a decremental gradient of antibiotic concentrations from plasma to the epithelial lining fluid and further to the exhaled breath condensate from HMEF. However, no statistical difference in antibiotic concentrations was observed between ELF and exhaled breath condensate collected from the HMEF. The mean antibiotic concentration in plasma was 293.5 (715) ng/mL, 12.3 (31) ng/mL in ELF, and 0.5 (0.9) ng/mL in the HMEF exhaled breath condensate. 2 = 0.02, p = 0.638), plasma and HMEF , or between ELF and HMEF .The antibiotic concentrations exhibited significant heterogeneity across the patient population, as depicted in In four HMEF samples, we detected very low antibiotic concentrations. Those samples induced a specific pattern that revealed the presence and identity of the antibiotic used. However, the values obtained were within the lower limit of detection (LOD). To account for this, we arbitrarily assigned a value of 0.01 ng/mL. Conversely, the three control HMEFs showed no signal when testing for PenPcf activity.In this exploratory study, we identified and quantified the concentration of beta-lactam antibiotics in the exhaled breath condensate collected from the HMEF from critically ill immunosuppressed patients. Moreover, we observed a decremental gradient of antibiotic concentrations from plasma to ELF and to HMEF exhaled breath condensate. However, we did not find any correlation between the three compartments assessed .In previous years, the study of EBC has experienced a notable increase in interest from researchers, primarily driven by the development of novel sampling methodologies and then by advances in analysis techniques. Numerous preclinical and clinical studies have been conducted to assess this matrix exploring various biomarkers in diverse respiratory conditions, including cancer , inflammn = 9), but similarly to us, they could not find a good correlation between the exhaled antibiotic concentration and the plasma antibiotic concentration [McNeil et al. studied EBC from HMEFs from critically ill patients. They suggested that there is a novel, non-invasive method to sample the distal airspace in patients with ARDS accurately. They found a good correlation between proteome isolated from undiluted pulmonary edema fluid and exhaled breath condensate collected from HMEF. In addition, they could discriminate between cardiogenic and inflammatory pulmonary edema based on differences in proteins identified by high-performance liquid chromatography (HPLC) . More rentration . There intration . The lack of correlation observed between antibiotic concentrations in different compartments, as reported in our study, can be attributed to various factors and underscores the complex pharmacokinetics of antibiotics within the lungs. Firstly, the plasma levels of beta-lactam antibiotics exhibit significant variability over time, characterized by fluctuating peaks and troughs, and varying ascending or descending slopes, influenced by factors such as clearance and specific drug properties . In our In addition, patients had received different antibiotic doses and for different periods when BAL was performed, so some patients were probably in the steady state while others were in the ascendant limb of the plasma time–concentration curve. Finally, the renal clearances and volumes of distribution varied between patients . These vELF antibiotic concentration depends on several factors such as the plasma concentration, protein binding, and physicochemical characteristics of the drug . HoweverAntibiotic deposition on the HMEF will result, in part, from an equilibrium between the plasma and ELF compartments. From the ELF, the antibiotic should be transported by tiny drops through airways up to the HMEF ; these d2 = 0.99) with concentrations determined by isocratic HPLC [The determination of beta-lactam antibiotics in ELF or HMEF using the PenPcf biosensor has not yet been validated using HPLC. Nevertheless, we previously demonstrated that the concentration of antibiotics in complex matrices such as plasma in the range between 130 and 6500 uM determined by the PenPcf biosensor correlate (Rtic HPLC . HoweverWe observed four HMEF fluid samples in which antibiotic concentrations were above the LLOD (lower limit of detection). However, they were below the lower limit of quantitation (LLOQ), where the amount of antibiotic can be confidently extrapolated. This issue has been extensively studied previously, and it has been shown that excluding the data between the LLOD and the LLOQ increases the bias and decreases the precision of the statistical analysis, especially when the percentage of data discarded is higher than 10%. Although some rules have been described for assigning values to these types of determinations, such as “LLOQ/2”, this rule can only be applied after a well-characterized statistical method ,24,25,26Despite the advances in the study of EBC, there are scarce data regarding the measurement of antibiotic concentrations in this matrix. We consider this an important issue because one of the main therapeutic objectives in the clinical setting is to achieve proper antibiotic concentrations at the infection site , avoiding systemic toxicity and adverse effects secondary to the diagnostic method and to the increasing bacterial resistance secondary to infra-therapeutic levels of antibiotic. To advance towards these objectives requires improving the knowledge of antibiotic pharmacokinetics in different biological compartments of critically ill patients. This information, in the future, could help clinicians to tailor the individual selection, dosage, and administration intervals of specific antibiotics. Thereby, future technological developments could monitor the adequate dosing titration of antibiotics for patients with pneumonia, similar to what happens today with inhaled anesthetics in the operating room ,46. We acknowledge some limitations in our study: first, we studied a small sample size. Second, we did not use the gold-standard method (HPLC) to measure antibiotic concentration; we consider that the use of HPLC may have been a good tool to measure the very low concentrations of antibiotics as we observed in HMEF exhaled breath condensate. Third, despite the common factor of respiratory failure in our patients, they presented a great heterogeneity in their clinical conditions. Fourth, we did not control the time of sampling. Sixth, we did not perform pharmacokinetic models or repeated measures to understand how the antibiotics were distributed across the different compartments over time. Finally, we cannot discard that the time of permanence of HMEFs could have influenced the antibiotic concentration measured.In conclusion, beta-lactam antibiotics can be detected and measured from the exhaled breath condensate accumulated in HMEFs. However, we found no correlations among the plasma, ELF, and HMEF exhaled breath condensates regarding the concentration of antibiotics. Future studies should try to better understand the factors involved in the transport of antibiotics from the alveoli to the HMEF to determine its potential as a novel source for the therapeutic drug monitoring of antibiotics.