Modulus Method Research Articles

Thermal-vibration ageing characteristics of three thin-walled cylindrical shells covered with a functionally graded protective coating: Modeling, analysis and test

In this study, the thermal-vibration ageing characteristics of three thin-walled cylindrical shells (TCSs) covered with a functionally graded protective coating (FGPC) are investigated for the first time, including fiber reinforced polymer (FRP), metal and fiber metal hybrid (FMH) shell structures with the FGPC. Initially, a dynamic model of an FGPC-TCS structure in a thermal-vibration ageing environment is developed by employing the first-order shear deformation principle enriched by the improved exponential function approach, the complex modulus method, the Rayleigh-Ritz method, etc. In this model, the ageing-related material parameters of the FGPC and the TCS structure are assumed to be affected by both environmental temperature and aging time. After the solution procedures of the vibration parameters of the above three coated shells are clarified, the determining method of fitting coefficients of the coating and different materials in the shells studied is provided. Also, the detailed experimental validation is undertaken on the uncoated and coated FRP, metal, and FMH shell specimens, which indicates that the present model is reliable for predicting the key dynamic ageing parameters when the complex thermal-vibration ageing effect is considered. From the measured and calculated data, it is proved that the suppression contribution of the FGPC is obvious no matter whether these shell structures are at room temperature or thermal degradation environment. Furthermore, some important application suggestions of the FGPC are summarized to improve the degradation resistance of such coated shell structures.

On the Plane Geometry Design Method of the Loop-free Hyperbolic Single-layer Cable Structures

The plane geometry design of the cable-net structure is the basis for the form-finding and structural design. The loop-free single-layer cable-net structure is a new cable-net scheme, but due to its special and complex shape, it is difficult to determine its plane geometry according to architectural requirements. In this paper, the moving support node method is proposed, which can quickly get the plane geometry of a loop-free single-layer cable-net structure. The form-finding is proceeded based on the existing small elastic modulus method. There are two constraints when using the moving support node method. One is that the projection of all cables is straight line, and the other is that the distance between the two cables of each support node and the inner boundary of the target is equal. The aim is to minimize the sum of the distances from all cables to the inner boundary of the target. Using the plane geometry control parameters and the plane coordinates of the support nodes as decision variables, the plane geometry closest to the target inner and outer boundaries is get through optimization analysis. According to the derivation of the formula, the plane projection shape of a loop-free single-layer cable-net structure is mainly related to three parameters, including the number of supports, the difference between the node numbers of the supports at both ends of each cable, and the target curve equations of the inner and outer boundaries. Finally, taking the roof structure of Linyi Olympic Stadium as an instance, the feasibility of the plane geometry design method of the loop-free single-layer cable-net structures based on the moving support node method is verified.

Water Inflow Amount Prediction for Karst Tunnel with Steady Seepage Conditions

Tunnel engineering is an important aspect of developing and utilizing underground spaces. Tunnel water inrush became a common problem that restricts the safe and efficient construction of tunnels. This paper focuses on a karst water-rich tunnel in Chongqing and establishes a seepage field distribution model around the tunnel, analyzing the evolution law of the seepage field. The water balance method and underground runoff modulus method are used to predict tunnel water inflow. The prediction method for tunnel water inflow in water-rich karst areas is combined with long-term on-site tunnel hydrology observations. The distribution of groundwater in front of the tunnel face is drawn using the software, successfully predicting the larger karst area in front of the face. The prediction of water inrush risk level for karst tunnels is carried out using the SVR model. An expression formula for the water head around the tunnel is established by using the conformal mapping relationship, and the distribution status of the seepage field around the tunnel is ultimately determined. The overall prediction accuracy of the underground runoff modulus method is better than that of the water balance method in predicting the water inrush volume of the tunnel. The prediction of the large karst area ahead of the heading is successfully achieved by using the SVR model. This prediction method can provide reference and guidance for the construction of other karst-rich water tunnels in the region.

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.

Accurate Detection of Concrete Pavement Thickness Based on Ultrasonic Array

Thickness detection of concrete pavement is a critical step in construction completion acceptance and serves as an important metric for subsequent pavement performance evaluations. The crux of thickness evaluation lies in determining the interface reflection echo propagation sound time. Based on the acoustic impedance difference between the surface layer and contact layer, pavement can be classified into two types: large-difference and small-difference. By examining the singularity of the detection signal, we employed the improved correlation coefficient method and the wavelet transform maximum value method to identify the interface reflection echo sound time. This study refines the traditional correlation coefficient method by using the surface direct wave as the reference signal, simplifying the signal analysis process. In actual detection, for the type of concrete pavement with large differences in acoustic impedance, the average relative error between the improved correlation coefficient method and the core drilling method can be reduced to 1.7%; for the type with small differences, the average relative error of the detection results obtained using the wavelet transform modulus maximum method can be reduced to 1.9%, showcasing high accuracy.

Experimental study on fatigue damage of self-compacting concrete of CRTS III slab track

To study fatigue performances and establish the fatigue damage model of self-compacting concrete (SCC) of China Railway Track System (CRTS) III slab track under different stress levels, static and fatigue tests were conducted. In the static test, the flexural strength of SCC was obtained. In the fatigue test, fifteen specimens were divided into five groups. The maximum stress levels were 0.65, 0.70, 0.75, 0.80, and 0.85 for each group, and the minimum stress level was 0.10 for every group. The S-N curve of SCC is obtained and shows that the fatigue life of SCC decreases with the increase of the maximum stress level. The strain evolution and elastic modulus degradation can be divided into 'Fast-Slow-Fast' three stages. With the maximum stress level increases, the proportion of elastic modulus degradation increases and decreases in the first and third stages, respectively. The fatigue damage model derived from the improved elastic modulus method and the Chaboche fatigue damage model can reflect the damage evolution of SCC under different stress levels. Moreover, the proposed model can provide theoretical guidance for the further numerical simulation of SCC for ballastless tracks under the fatigue load.

Theoretical prediction method of Young's modulus and yield strength of micron particle reinforced metal matrix composites at different temperatures

To begin with, a theoretical characterization model of temperature and porosity dependent Young's modulus for micron particle reinforced metal matrix composites is established by considering the evolution of properties of reinforced particles and metal matrix with temperature. Additionally, combining the existing strengthening mechanism theory and incorporating the influence of grain boundary slip on the related strengthening mechanism and the yield strength of metal matrix, a temperature dependent yield strength analysis model of micron particle reinforced metal matrix composites is proposed. These models only require material parameters at room temperature and temperature dependent specific heat capacity at constant pressure for application. In addition, the predicted results from these models are reasonable and consistent with measured results. Moreover, based on the established models, the effects of key material parameters and main mechanisms on Young's modulus and yield strength of composites and their variation with temperature are explored. Furthermore, the variation of various control mechanisms with the particle size and temperature is clarified. It lays a theoretical foundation for the development of micron particle reinforced metal matrix composites that are suitable for high-temperature environments and for the optimization of their key parameters.

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.

A Prediction Method of Effective Modulus of POSS Modified Quartz Fiber Reinforced Silicone Matrix Unidirectional Composites

In the manufacturing process of Polyhedral oligosilsesquioxane(POSS) modified quartz fiber reinforced silicone resin composite, if the POSS content ratio and the combination performance of POSS and silicone resin can be effectively controlled, the comprehensive performance of the material can be significantly improved, and many weaknesses of the composite can be improved. At the same time, the substantial problems such as the stiffness and strength of the wave transmitting composite still need to be greatly improved can be solved, and the new wave transmitting composite can be widely used in the main bearing parts of aerospace. When POSS modified quartz fiber reinforced silicone resin matrix composites are used as structural materials, their mechanical properties, such as modulus, are one of the most important macro mechanical material parameters in engineering applications. However, there is no effective prediction method in engineering to give practical guidance. Therefore, this paper establishes a prediction method of effective modulus of POSS modified quartz fiber reinforced silicone resin matrix composites based on micromechanics, The system can accurately understand the relationship between material structure and properties, thus making it possible to predict properties and design materials, and quickly guiding the practical application of engineering.

Nonlinear Settlement Calculation of Composite Foundation Based on Tangent Modulus Method: Two Case Studies

The tangent modulus method of undisturbed soil is a new method in settlement calculation, which is mainly applied to hard soil with a strong structure, such as silty clay, completely weathered rock, and granite residual soil with an SPT blow count greater than 8. The tangent modulus is mainly obtained from a field plate load test, which can consider the influence of the soil stress level and reflect the nonlinear characteristics of the foundation settlement. In a multi-layer soil foundation, since the deep plate loading test is difficult, a method was proposed to determine the tangent modulus of deep soil. It is assumed that the ratio of the initial tangent modulus to the deformation modulus is equal to the ratio of the unloading–reloading modulus Eurref to the secant modulus E50ref obtained by triaxial unloading–reloading test. Since there are corresponding empirical formulae for SPT counts and the deformation modulus of different types of soils in many regions, the initial tangent modulus can be derived by the above method. In two cases of a composite foundation, the compression modulus and tangent modulus were used to calculate the settlement of the foundation, which is then compared with the measured results. The results show that the proposed method for determining the tangent modulus of deep soil is feasible in theory, and the calculating accuracy of the tangent modulus is significantly higher than that of the traditional compression modulus.

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.

Bond-slip effect in novel steel box-concrete composite structures part Ⅱ: Long-term loading

Considering bond-slip effect, the structural response of steel box-concrete composite beams (SBCCB) subjected to short-term load has been investigated utilizing theoretical analysis, numerical simulation, and loading test in PartⅠof this study. In PartⅡof this study, the effect of shrinkage and creep of concrete core will be quantitatively studied to investigate the long-term response of SBCCB. For analyzing long-term structural performance, equivalent elastic modulus method is introduced to deduce the theoretical formulas of deflection and stress. Besides, the long-term deflection and stress considering bond-slip effect are also derived. In addition, finite element models are constructed to further validate the accuracy of theoretical solutions. Creep coefficient and shrinkage strain determined according to six kinds of specifications are compared. Then, comparative analysis of long-term stress and deflection, as well as the comparison of numerical and theoretical results, are performed. Research findings show that the proposed formulas are applicable for predicting long-term response of SBCCB. Meanwhile, the additional stress and deflection induced by creep and shrinkage of concrete core in SBCCB are not neglectable and should attract enough attention. Increment ratios corresponding to additional deflection and stress are 20%-25% and 8–10%, respectively. Combined with PartⅠof this study which focuses on short-term performance, the research findings in Part Ⅱ can further improve the long-term design methodology of practical SBCCB.

Properties of the Quadratic Transformation of Dual Variables

We investigate a solution of a convex programming problem with a strongly convex objective function based on the dual approach. A dual optimization problem has constraints on the positivity of variables. We study the methods and properties of transformations of dual variables that enable us to obtain an unconstrained optimization problem. We investigate the previously known method of transforming the components of dual variables in the form of their modulus (modulus method). We show that in the case of using the modulus method, the degree of the degeneracy of the function increases as it approaches the optimal point. Taking into account the ambiguity of the gradient in the boundary regions of the sign change of the new dual function variables and the increase in the degree of the function degeneracy, we need to use relaxation subgradient methods (RSM) that are difficult to implement and that can solve non-smooth non-convex optimization problems with a high degree of elongation of level surfaces. We propose to use the transformation of the components of dual variables in the form of their square (quadratic method). We prove that the transformed dual function has a Lipschitz gradient with a quadratic method of transformation. This enables us to use efficient gradient methods to find the extremum. The above properties are confirmed by a computational experiment. With a quadratic transformation compared to a modulus transformation, it is possible to obtain a solution of the problem by relaxation subgradient methods and smooth function minimization methods (conjugate gradient method and quasi-Newtonian method) with higher accuracy and lower computational costs. The noted transformations of dual variables were used in the program module for calculating the maximum permissible emissions of enterprises (MPE) of the software package for environmental monitoring of atmospheric air (ERA-AIR).

Research on Mechanical Properties of Winding Axial Bending Under Unbalance of Transformer Pad

To improve the study on the influence of transformer windings and pad support on transformer short-circuit resistance, the calculation method of wire bending elastic modulus is analyzed, and the sample test is carried out: The influences of the unbalance of the two ends of the wire between the number of wires wound and the span of adjacent pads on the force-deflection curve, the bending force at the selected deflection, and the elastic modulus of bending were analyzed. The results showed that: Increasing the number of wires and decreasing the height difference of adjacent pads can improve the bending force and bending elastic modulus of wires, the axial stability, and short-circuit resistance of transformer windings. When the height of the adjacent pad is not balanced, the axial force can increase the compression degree between the pad and the wire, and reduce the span of the wire. Under the action of force, the wire yields twice, which further reduces the bending force and elastic modulus of the wire. This research can provide test data support for improving the installation of transformer winding pads and theoretical evaluation of short-circuit tolerance.

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.

Nonlinear vibration characteristics of composite pyramidal truss sandwich cylindrical shell panels with amplitude dependence

Recently studies on the bending, compression, buckling, and other static issues of composite pyramidal truss sandwich cylindrical shell (CPTSCS) panels have been reported. However, rare literature has explored the nonlinear vibration characteristics of the CPTSCS panels. In this paper, their amplitude-dependent vibration properties are studied both experimentally and theoretically. First, a fabrication flowchart is proposed to obtain such panel specimens, and a pulse excitation method via a hammer and sweeping frequency and fixed excitation techniques provided by a base vibration shaker are employed to measure the nonlinear variation of vibration parameters. As a result, the amplitude-dependent vibration phenomenon is confirmed to exist in both the pyramidal truss core and composite skins. After the nonlinear assumptions of Young's moduli, shear moduli, and loss factors of the pyramidal truss core and composite skins are defined based on Jones-Nelson nonlinear material theory, complex modulus method, Gibson equivalent principle, etc., a novel theoretical model is developed to predict the nonlinear natural frequencies, damping ratios, and resonant responses under different base excitation energies, in which the first order shear deformation theory, Rayleigh Ritz variation method, modal strain energy approach, and modal superposition technique are adopted to solve key vibration parameters. The pre-defined amplitude-dependent fitting coefficients that are vital to the completeness of the current model are further identified using the measured frequency and damping data. Also, detailed tests are done to make sure that both the proposed model and the predictions are correct. Finally, the effects of critical geometric and material parameters of the CPTSCS panels on the amplitude-dependent vibration characteristics are evaluated based on the calculated dynamic results, with some practical suggestions being refined for improving the vibration suppression capability of such advanced composite structures.

A characterization method for equivalent elastic modulus of rock based on elastic strain energy

Energy is an internal variable during rock deformation and failure, and its dissipation and conversion law can reflect the rock’s internal damage and deterioration state. Analysis of rock deformation and failure process from the perspective of energy is helpful to deeply understand the mechanism of rock damage, fracture and instability failure, and has important theoretical and practical significance for the stability evaluation and support control of surrounding rock. In this study, through single cyclic loading and unloading (SCLU) experiments, cyclic triaxial loading and unloading (CTLU) experiments and conventional triaxial compression (CTC) experiments, the equivalent elastic modulus method based on elastic strain energy is proposed to analyze the energy conversion of rock. The results show that the error of the elastic strain energy calculated by the strain energy formula method is generally higher than 10% with the secant and tangent modulus of the loading and unloading curve as input parameters. Taking the equivalent elastic modulus proposed in this study as an input parameter, more accurate elastic strain energy can be obtained by the strain energy formula. During the rock failure process, the equivalent elastic modulus shows a three-stage characteristic of increase, steady and decrease. The equivalent elastic modulus can be estimated by the quadratic function between the equivalent elastic modulus and confining pressure and axial strain. Under the same deformation and deviatoric stress, the elastic strain energy stored in rock increases with increasing confining pressure. The local maximum energy dissipation rate corresponds to stress drop, and the peak energy dissipation rate appears near the peak strength. High energy dissipation mainly occurs in a short time after peak strength, and energy release and dissipation are more sudden and severe under high confining pressure.

Study on Calculation Method of Foundation Nonlinear Settlement Based on In Situ Loading Test

In this paper, a nonlinear deformation modulus method is proposed for foundation settlement calculation. In the proposed method, the nonlinear deformation modulus under different stress levels is obtained from the load-settlement curve of in situ loading test, which are then applied to the layerwise summation method for calculating foundation settlement. On this basis and referring to the Duncan-Chang model, a variable modulus constitutive model suitable for numerical calculation of foundation settlement is further proposed. The required parameters of this model are the same as those of the nonlinear deformation modulus method and can be determined by the in situ loading test. The validity of the proposed calculation methods of foundation settlement is verified by the in situ loading tests under different plate sizes. The results illustrate that both the nonlinear deformation modulus method and the variable modulus constitutive model compare quite well with the test results, and the deduced results can better reflect the nonlinearity of foundation settlement.