Effective Modulus Method Research Articles

A Numerical Study on the Influence of Bolt Corrosion on the Long-Term Behavior of Steel–Concrete Composite Beams

This study establishes a numerical model for beam-type steel–concrete composite specimens considering the corrosion of anchor bolts, with which studies the long-term deformation performance of the components under sustained load. The experimental results were compared with the calculated values obtained by combining the effective modulus method according to the CEB-FIP 1990 code and the ACI 209R code. Then a comparison with the calculation method of shrinkage and creep in standards (Standard Creep Method (SCM)) is made, and it shows that by supplementing the degradation of interface stiffness due to corrosion (using the defined modulus method (DMM)), the simulation results match better with the experimental results, confirming that this approach is suitable for analyzing the long-term load conditions of anchor bolt corrosion. Furthermore, based on the defined modulus method, the long-term behavior of composite beams under different loads and varying corrosion rates is studied.

EFFECTIVE MODULUS METHOD (EMM) CONCEPT APPLIED TO THIN‐WALLED STEEL BEAMS AFFECTED BY LOCAL‐DISTORTIONAL INTERACTION

AbstractThe Effective Modulus Method (EMM), which simulates the effects of local and distortional post‐buckling to establish simplified design methodologies concept, is applied to the determination of the beam strength. This methodology was presented before [1], where the authors tested the results in columns, in comparison with experimental tests. In this approach, local and distortional post‐buckling effects are simulated by changing the mechanical characteristics (namely the elasticity modulus) of the cross‐section of the members. Specifically, the paper presents (i) the proposal of a new design methodology for beams affected by local and distortional buckling modes interaction and (ii) the comparison with experimental tests available in the literature.

Effective Modulus Method (EMM) concept applied to thin‐walled steel columns.

AbstractThe design of thin‐walled steel members, subjected to compressive stresses, requires complex analysis to account the interaction between local, distortional and global instabilities. There are simplified methodologies that allow simulating the effects of local and distortional post‐buckling behavior in usual structural members. The Effective Width Method (EWM) performs this simulation by changing the geometry of the cross‐section, being the pioneer methodology for the analysis of thin‐walled members. The Direct Strength Method (DSM) allows the determination of resistant forces through simple‐to‐use formulas, eliminating the calculation of the geometric properties of the section as required by the EWM. This paper presents the Effective Modulus Method (EMM) concept, which simulates the effects of local and distortional post‐buckling, to establish simplified design methodologies, desirably leading to more rigorous results than those provided by the available methodologies. In this approach, local and distortional post‐buckling effects are simulated by changing the mechanical characteristics (namely the modulus of elasticity) of the cross‐section of the members. Specifically, the article presents (i) the proposal of the new design methodology and (ii) the comparison with experimental and numerical results obtained by finite element shell models.

A comparative study for determining rock joint normal stiffness with destructive uniaxial compression and nondestructive ultrasonic wave testing

Rock joint a kind of vital discontinuities in natural rock mass. How to accurately and conveniently determine joint normal stiffness is therefore significant in rock mechanics. Here, first, seven existing methods for determining joint normal stiffness were introduced and reviewed, among which Method I (the indirect measurement method), Method II (the direct determination method), Method III (the across-joint strain gauge measurement method) and Method IV (the deformation measuring ring method) are via destructive uniaxial compression testing, while Method V (the best fitting method), Method VI (the rapid evaluation method) and Method VII (the effective modulus method) are through wave propagation principles and nondestructive ultrasonic testing. Subsequently, laboratory tests of intact and jointed sandstone specimens were conducted following the testing requirements and procedures of those seven methods. A comparison among those methods was then performed. The results show that Method I, i.e. the benchmark method, is reliable and stable. Method II has a conceptual drawback, and its accuracy is acceptable at only very low stress levels. Relative errors in the results from Method III are very large. With Method IV, the testing results are sufficiently accurate despite the strict testing environment and complicated testing procedures. The results from Method V are greatly unstable and significantly dependent on the natural frequency of the transducers. The joint normal stiffness determined with Method VI is stable and accurate, although data processing is complex. Method VII could be adopted to determine the joint normal stiffness corresponding to the rock elastic deformation phase only. Consequently, it is suggested that Methods I, IV and VI should be adopted for the measurement of joint normal stiffness. The findings could be helpful in selecting an appropriate method to determine joint normal stiffness and, hence, to better solve discontinuous rock mass problems.

Study on the Mechanical Properties of Continuous Composite Beams under Coupled Slip and Creep

Steel–concrete continuous composite beams are widely used in buildings and bridges and have many economic benefits. Slip has always existed in composite beams and will reduce the stiffness of composite beams. The effect of creep under a long-term load will also be harmful. Many scholars ignore the combined effects of slip and creep. In order to more accurately study the mechanical properties of steel–concrete continuous composite beams under long-term loads, this paper will consider the combined actions of slip and creep. By combining the elastic theory and the age-adjusted effective modulus method, the differential equation of the composite beam is derived via the energy variational method. The analytical solutions of axial force, deflection and slip under a uniform load are obtained by substituting the relevant boundary conditions. The creep equation is used to simulate the behavior of concrete with time in ANSYS. The analytical solution is verified by establishing a finite element model of continuous composite beams considering slip and creep. The results suggest the following: the analytical solution is consistent with the finite element simulation results, which verifies the correctness of the analytical solution. Considering the slip and creep effects will increase the deflection of the composite beam and the bending moment of the steel beam, reduce the bending moment of the concrete slab and have a significant impact on the structural performance of the continuous composite beam. The research results considering the coupling effect of slip and creep on continuous composite beams can provide a theoretical basis for related problems.

A general equation for estimation of time-dependent prestress losses in prestressed concrete members

Due to the inherent nature of concrete shrinkage, creep and relaxation of prestressing strands, the prestress in prestressed concrete structures decreases continuously with time, i.e., there is an apparent time-dependent prestress loss phenomenon for prestressed concrete structures. Therefore, it is an important issue for the estimation of time-dependent prestress losses in the design of prestressed concrete structures. Many current design codes provide relevant equations for the evaluation of prestress losses. For the prestressed concrete structure, in addition to the prestressing strands, a great amount of ordinary reinforcing steel is configured to enhance its load-carrying capacity. It is generally considered that no stress relaxation occurs for ordinary reinforcing steel due to its lower stress. However, the presence of ordinary reinforcing steel affects the stress distribution in the structure and thus has an obvious impact on the prestress losses, which is not considered in the prediction equations in most current codes for the design of concrete structures up to date. In this study, based on the cross-sectional analysis of prestressed concrete members that combines the age-adjusted effective modulus method of concrete creep analysis, a general equation for estimation of the time-dependent prestress losses in prestressed concrete members considering the influence of ordinary reinforcing steel is derived and simplified. When the influence of ordinary reinforcing steel is ignored, i.e., the percentage of ordinary reinforcing steel is taken as zero, the proposed equation reduces to those in Eurocode EN 1992-1-1:2004 and AASHTO LRFD 2020. In the end, an example of a T-shaped prestressed concrete girder is provided to show the application of the proposed equation for the evaluation of time-dependent prestress losses and the error resulting from the simplification of the equation. The results show that ordinary reinforcing steel has a significant effect on the time-dependent prestress losses in prestressed concrete members for the commonly used percentage of ordinary reinforcing steel, and the error arising from the simplification of the first derived equation is quite small.

Analysis of time-varying mechanical properties of prestressed concrete containment during the tensioning process and service considering the influence of creep

The containment structure is an important barrier that ensures the safe operation of nuclear power plants. Currently, most studies on concrete creep in containment structures assumed constant stresses in the calculation of creep, which is not consistent with realistic cases or circumstances. In this study, a combination of theoretical analysis and numerical simulation was used to investigate the prestressed concrete containment (PCC) of nuclear power plants. Based on the age-adjusted effective modulus method, three widely accepted concrete creep models were improved to obtain a variable stress creep model. Using the improved creep model, the effect of concrete creep on the PCC structure during prestressed tensioning and long-term service was systematically analyzed. Then, the internal pressure bearing capacity of the PCC in long-term service was assessed, and the effects of long-term concrete creep on the mechanical properties of the PCC were evaluated.

Experimental study and analysis for the long-term behavior of the steel–concrete composite girder bridge

Due to the influence of the meteorological factors, the long-term behaviors of steel–concrete composite girders of the real bridge is different from that of indoor laboratory models. In order to address this issue, a long-term field test of 4 years was performed on the simply-supported steel–concrete composite girder bridge to study the temperature, shrinkage and creep effects. The temperature field models (TFM) and mechanical behavior models (MBM) were built in sequence and verified by the test results of the concrete deck and steel girders of 2018 and 2019 since the temperature and strain changed greatly during this period. The temperature calculated by the TFM agreed well with the test results. Although the stress of composite girders obtained by the test is different from that of the MBM, the difference in the whole service stage is small. The test results of the steel cross beam, the steel–concrete interface, the wet joint and the concrete continuous layer were also discussed, and the former three were all under compressive state, while the concrete continuous layer is under tensile state in some time period with the maximum tensile stress 3.12 MPa. Then, the verified models were used to separate the temperature strain from the long-term tested strain, and the remaining tested strain is the shrinkage and creep strain. In summer and winter, the shrinkage and creep stress calculated by the age-adjusted effective modulus method (AEMM) was different from the tested values due to the influence of air humidity, initial load age, stress level and inherent error of the prediction model. Much attention should be paid to the influence of the meteorological factors on the long-term behaviors of steel–concrete composite girder bridges.

An Analytical Model to Evaluate Short- and Long-Term Performances of Post-Tensioned Concrete Box-Girder Bridges Rehabilitated by an Ultrahigh-Performance Concrete Overlay

Potentially unfavorable stress redistributions could occur in post-tensioned concrete box-girder bridges upon deck rehabilitation by an ultrahigh-performance concrete (UHPC) overlay. The reason for such redistributions is that the deck forms an integral part of the load-resisting mechanism so that deck rehabilitation, namely, the process of removing the deteriorated deck and casting a fresh UHPC overlay, may cause large stress redistributions in the box girder. In addition, the significantly different shrinkage behavior between the freshly cast UHPC deck overlay and the old existing concrete deck below would result in restrained stress in the UHPC–old concrete interface. Therefore, how to evaluate the stress redistributions and the potentially unfavorable stresses in the box girder resulting from construction operations and the incompatible shrinkage is a crucial issue. Here, we propose an analytical model to assess the short- and long-term performances of the concrete box girder during and after deck rehabilitation. The analytical model was verified by using finite-element models incorporating the staged construction simulation and time-dependent behavior of materials. The model can be easily established and calculated with a great amount of time-saving compared with finite-element modeling, and it applies to different types of post-tensioned concrete box-girder bridges. The analytical model involves the balancing-load concept and the age-adjusted effective modulus method (AEMM). The balancing-load concept in the short-term analysis is to account for the prestressing force, and the AEMM in the time-dependent analysis is an effective tool to determine how stresses and strains vary with time due to the creep and shrinkage of a UHPC deck overlay. A parametric study was then conducted based on the analytical model, which involved the thickness of the deck layer to be rehabilitated and the addition of steel reinforcement in the new UHPC deck overlay. Analysis results indicated that pronounced tensile stresses developed in the UHPC deck overlay, which resulted from the restraint of the free shrinkage of the UHPC overlay by the subjacent concrete deck. An increase in the thickness of the newly cast UHPC deck overlay resulted in a decrease of the tensile stress in the UHPC deck overlay, and a rehabilitation thickness of 3.8–5.1 cm (1.5–2.0 in.) was deemed appropriate for the selected bridge according to the analysis.

A step-by-step method for time-dependent analysis of composite beams

Composite beams composed of reinforced concrete slab and steel section joined by mechanical connectors. The concrete slab provides compressive strength, fire resistance, and floor surface, whereas the steel section provides high tensile strength. Steel-concrete composite structures are widely used in frame buildings and bridges due to their economic and structural advantages. In the existing literature, a hybrid analytical-numerical procedure is available for service load analysis of steel-concrete composite beams considering cracking, creep, and shrinkage in concrete. The procedure uses an age-adjusted effective modulus method (AEMM) to model creep and shrinkage effects. The AEMM is computationally efficient however, it compromises accuracy owing to the use of the same aging coefficient for creep and shrinkage. Therefore, in this paper, a step-by-step method (SSM) has been proposed for modeling the creep and shrinkage effects. The method takes into account the progressive concrete cracking with time due to creep and shrinkage. The method is computationally efficient for large composite frame buildings compared to the finite element method since no discretization of the member, along the length and/or across the cross-section, is needed. The method has been validated against experimental results available in the literature. It is observed that the SSM has greater accuracy than the AEMM.

Analytical investigation of long-term behavior of normal concrete filled UHPC tube composite column

The long-term creep and shrinkage behavior of a recently proposed normal concrete (NC) filled ultra-high-performance concrete (UHPC) tube (abbreviated as NFUT) composite column has been rarely investigated. The present study aims to propose an analytical solution for calculating the restrained stress and initial axial stress of NC core based on elastic mechanics knowledge. The nominal creep coefficient or strain of the NFUT composite column is analytically calculated as well, based on the age-adjusted effective modulus method. The perfect bond between the UHPC and NC is assumed and the shrinkage of the UHPC tube subjected to steam curing is negligible during the calculation. Moreover, a finite element model is established, and the numerical restrained stress and initial axial stress are compared with the analytical ones. Also, the effect of the Poisson’s ratio and reinforcement ratio in the UHPC tube on these stresses is investigated. The analytical nominal creep coefficients or strains of the specimens are compared with the test ones and the previous numerical parametric analysis results. It is found that the numerical restrained stress and initial axial stress show good agreement with the analytical ones with errors smaller than 10 % The restrained stress factor and initial axial stress are linearly proportional with the Poisson’s ratio difference between the UHPC and NC. Also, the analytical nominal creep coefficients generally show good agreement with the test ones, and the errors between the analytical and the numerical final strain predictions are within 1 %, which demonstrates the applicability and accuracy of the age-adjusted effective modulus method based nominal creep coefficient calculation.

Time-dependent behavior of NSM CFRP-strengthened RC beams under different service temperatures

The use of fiber reinforced polymer (FRP) for flexural strengthening of reinforced concrete (RC) beams has become a popular strengthening technique. Significant amount of work is available on the short-term flexural behavior of RC beams strengthened with near-surface mounted (NSM) technique. However, their time-dependent flexural behavior, specifically under high service temperature, has not yet been addressed. This paper presents an experimental work to evaluate the time-dependent behavior of NSM carbon FRP (CFRP)-strengthened RC beams. The experimental program included 23 beams, where the effect of different parameters such as strengthening (CFRP) area, steel reinforcement ratio and applied temperature (20 and 50 °C) have been considered. Experimental results show that the effect of strengthening area is significant on the flexural short-term response of the beams, while minor effects are found on the time-dependent deflections. On the other hand, increasing the service temperature has no significant effect on the short-term tests, but it produces a large increase in the time-dependent deflection of the specimens. Finally, an analytical procedure for the prediction of time-dependent deflections, which is based on the age-adjusted effective modulus method (AEMM), is presented. Good agreement between the experimental results and analytical predictions on time-dependent deflections is shown.

Long-term mechanical behaviors of uncracked concrete beams prestressed with external basalt fiber-reinforced polymer tendons

In this study, an experiment was conducted on four uncracked concrete beams prestressed with external basalt fiber-reinforced polymer (BFRP) tendons or stress-relieved strands subjected to 300-day sustained loads. The experimental variables included tendon type, prestress level (40 kN and 50 kN), and concrete strength grade (C40 and C60). The effects of these variables on the long-term deformation and prestress loss of the beams were discussed and analyzed. Based on an existing approach using an age-adjusted effective modulus method (AEMM), a modified model was proposed for predicting the long-term deformation and prestress loss of concrete beams prestressed with external BFRP tendons. The results indicated that the prestress level had significant effects on the long-term deformation and prestress loss. The concrete strength grade significantly affected the long-term deformation, but had no effect on the prestress loss. The stress-relieved strands exhibited a prestress loss that was 16% greater than that of the BFRP tendons. The theoretical results based on the modified model were consistent with the experimental results. The results of this study contribute to the design and prediction of the long-term behaviors of concrete beams prestressed with external BFRP tendons.

Research on vertical deformation and predeformation control of three‐tower connected super high‐rise structure during construction

SummaryConnected super high‐rise structure has a long construction period and significant time‐dependent characteristics, so it is necessary to study its vertical deformation during construction. Taking an asymmetrical three‐tower conjoined super high‐rise structure as engineering background, the equivalent elasticity problem was analyzed by age‐adjusted effective modulus method, and vertical deformation formula was deduced. A simplified model and a refined model of structure were established. The correctness of the deformation formula and modeling had been proven by comparing the dynamic characteristics and numerical simulation results. Based on the simplified model, the optimal construction plan of the conjoined super high‐rise structure was discussed. Based on the deformation formula, the predeformation calculation method of the conjoined super high rise was studied, and the correctness was verified by numerical simulation. The results show that the largest vertical deformation of the connected structure occurs at the position of corridor, and an apparent mutation occurs here. Properly delaying the rigid connection timing between corridor and tower is beneficial to the control of the overall structural deformation difference and internal force. The predeformation analysis method proposed in this paper is correct and effective, which could offset the vertical deformation of the structure during the construction sequence.

Experimental and numerical study on the in-plane behaviour of concrete-filled steel tubular arches with long-term effects

Concrete-filled steel tubular arch (CFSTA) bridges have been widely used in the past few decades. As the service time of the arch rib increases, the mechanical performance degrades due to the time-deterioration properties of infilled concrete. In order to study the influence of the long-term effects on the ultimate bearing capacity of arch ribs, four CFSTAs totally with two different fly ash replacement ratios were designed and tested to failure after a one-year sustained loading test. Test results showed that the in-plane bearing capacity of the post-creep CFSTAs was obviously reduced compared to that without long-term loading cases. A stepwise approach for considering the long-term behaviour of concrete was elaborated on and numerically implemented in ABAQUS, and the proposed finite element (FE) model was verified against the results yielded from the experiment. The study found that the age-adjusted effective modulus method may not accurately capture the in-plane behaviour of the post-creep coupons. FE results showed that the influence level of long-term effects on the capacity was negatively correlated with the stiffness of the arch rib. For CFSTA bridges within engineering applications, the capacity reduction caused by shrinkage and creep effects can be more than 10% after ten years.

Numerical simulation of concrete creep behaviour using integral creep algorithm with alternating stresses

To investigate the accuracy of commonly used algorithms for prediction of creep and shrinkage effects in concrete structures, a comparative study is carried out using the finite element (FE) models with various user-defined time-dependent material properties. Numerical results are compared with available benchmark tests to justify the FE models. Results show that the FE model following the concrete integral creep (CIC) algorithm is close to the test result. The secondarily developed subroutine on the universal commercial FE platform is able to represent the time-dependent behavior of the concrete cylinders. CIC shows higher accuracy once alternating load history is considered, though the rate of creep method (RCM) and age-adjusted effective modulus method (AEMM) show the same accuracy as CIC under constant load. CIC shows outstanding compatibility with the material model specified in contemporary design codes. It seems that the CIC model can be used for a detailed evaluation of concrete structures, while RCM and AEMM might be used for preliminary and rough analysis.

A Software Tool to Estimate the Time-dependent Deformation in RC Walls and Columns

Tall reinforced concrete (RC) buildings are sensitive to time-dependent effects which arise due to creep and shrinkage. In this study, a software tool has been developed to estimate the time-dependent deformations in RC walls and columns. The program can invoke the ACI 209, EC 2, IS 1343 and fib MC 10 models to determine the creep and shrinkage in plain concrete. In order to include the effects of longitudinal reinforcement, two alternative methods have been explored: (i) effective modulus method and (ii) one-dimensional boundary value problem method. The effect of variable environment is included using the principle of superposition, which is valid within the linear creep domain. The program is written on the Python platform and can be used on any Windows machine. The program is validated using tests reported in the literature. The program is capable of accepting input of a linear elastic one-step-analysis from standard commercial software like ETABS and StaadPro. This program requires minimal input from the user, such as the grade of concrete, rate of loading and time at which the deformations are to be evaluated. The time-dependent deformations are evaluated for each column or wall and the super-elevation levels for each story are presented in a chart form, which serves as a design aid for the engineers and can be readily used by the site construction team.

Finite Beam Element for Curved Steel-Concrete Composite Box Beams Considering Time-Dependent Effect.

Curved steel–concrete composite box beams are widely used in urban overpasses and ramp bridges. In contrast to straight composite beams, curved composite box beams exhibit complex mechanical behavior with bending–torsion coupling, including constrained torsion, distortion, and interfacial biaxial slip. The shear-lag effect and curvature variation in the radial direction should be taken into account when the beam is sufficiently wide. Additionally, long-term deflection has been observed in curved composite box beams due to the shrinkage and creep effects of the concrete slab. In this paper, an equilibrium equation for a theoretical model of curved composite box beams is proposed according to the virtual work principle. The finite element method is adopted to obtain the element stiffness matrix and nodal load matrix. The age-adjusted effective modulus method is introduced to address the concrete creep effects. This 26-DOF finite beam element model is able to simulate the constrained torsion, distortion, interfacial biaxial slip, shear lag, and time-dependent effects of curved composite box beams and account for curvature variation in the radial direction. An elaborate finite element model of a typical curved composite box beam is established. The correctness and applicability of the proposed finite beam element model is verified by comparing the results from the proposed beam element model to those from the elaborate finite element model. The proposed beam element model is used to analyze the long-term behavior of curved composite box beams. The analysis shows that significant changes in the displacement, stress and shear-lag coefficient occur in the curved composite beams within the first year of loading, after which the variation tendency becomes gradual. Moreover, increases in the central angle and shear connection stiffness both reduce the change rates of displacement and stress with respect to time.

Ageing coefficient for early age tensile creep of blended slag and low calcium fly ash geopolymer concrete

The ageing coefficient is required in the age-adjusted effective modulus method to assess the effect of creep in concrete structures subjected to varied stress history. In this paper, experiments were carried out to calibrate the ageing coefficient for early age tensile creep of blended slag and low calcium fly ash geopolymer concrete. The development of total strains of geopolymer concrete under sustained tension, including instantaneous strain, creep strain and shrinkage strain was monitored by using the dog-bone shaped specimens. The specimens were loaded at the age of 2, 3, 4, 7, 14, 21 days, respectively. The strains of the unloaded companion specimens were monitored as well. The development of the creep coefficient φ(t,τ0) for geopolymer concrete was calculated based on the experimental results. By using the step-by-step numerical analysis, the ageing coefficient for early age tensile creep of geopolymer concrete was assessed. An ageing coefficient of 0.8 is recommended for structural design. The comparison between calculated and measured tensile strains from the restrained concrete ring test shows the validity of the proposed value of the ageing coefficient.

Long-term performance of prestressed concrete continuous box girders in the natural environment

A long-term load test performed for 470 days on two two-span prestressed concrete (PC) continuous box girders is reported in this paper. Load types were selected as the test variates, and structural responses such as support reactions, deflections, and concrete strains were monitored. Simultaneously, affiliated experiments such as material strength, creep, and shrinkage tests were conducted to investigate the time-dependent performances of the materials. Data obtained from these tests showed that deflections, strains, and support reactions develop rapidly in the beginning and stabilize afterward; the reactions of mid- and end-supports decline and rise over time, respectively. Time-dependent patterns of deflections and support reactions were analyzed on the basis of an effective modulus method, and a practical calculation method for long-term deflections considering reaction redistributions was proposed. The effects of the service environment on the performance of PC girders were evaluated through an incremental analysis method.