A three-dimensional cyclic symmetry finite element model of titanium-matrix composites(TMCs) ring was developed to investigate the stress distribution and burst failure. The effects of fiber volume fractions, reinfo...A three-dimensional cyclic symmetry finite element model of titanium-matrix composites(TMCs) ring was developed to investigate the stress distribution and burst failure. The effects of fiber volume fractions, reinforced areas, thermal residual stresses and two different temperatures on stress distribution were studied. The burst speed was obtained through analyzing the hoop tensile stresses under a series of rotating speeds. The results indicate that at the two different temperatures, the influences of fiber volume fractions and reinforced areas on stress level and distribution are different. Some proposals are provided for the structure design of the TMCs ring. With regard to thermal residual stresses, a larger reinforced area is an advisable choice for design of the ring at higher temperature.展开更多
This paper presents finite element modeling (FEM) to simulate the present-day stress field and crustal deformation using NE-SW structural section in the central Seismic Gap region of the Garhwal Himalaya. Our study ...This paper presents finite element modeling (FEM) to simulate the present-day stress field and crustal deformation using NE-SW structural section in the central Seismic Gap region of the Garhwal Himalaya. Our study deals with the effect of geometrical characteristics and rock layer parameters on the upper crust. Modeling results show that two types of tectonic regimes developed in the central Seismic Gap region: the geotectonics of the northern part has been controlled by regional compression, whereas southern part is characterized by regional extension. Correspondingly, thrust faults are induced in the northern part and normal faults are extensively developed in the southern front. Those evidences noticeably indicate that the compressive tectonic environment of the Himalaya becomes change into the extensional tectonic regime in its front. The computed shear stress accumulation along the northern fiat of Main Himalayan Thrust (MHT) implies that considerable amount of interseismic stress is building up along the MHT system in the Himalaya, which ultimately release through the possible future great Himalayan earthquake (M 〉 8). The comparison between our modeled stress field, faulting pattern and horizontal shortening rate with the distribution of the microseismic events, focal mechanism solutions, active faulting and GPS data in the central Seismic Gap region shows good agreement.展开更多
Deformation behavior of slab at the straightening stage during continuous casting was simulated by the explicit dynamic finite element method,and the stress distribution along the width direction of the slab and its c...Deformation behavior of slab at the straightening stage during continuous casting was simulated by the explicit dynamic finite element method,and the stress distribution along the width direction of the slab and its change regularity at slab center during continuous casting were obtained.The influence of distribution and change of stress on the propagation of longitudinal cracks on slab surface was discussed.The results show that the tensional stress appears on slab surface at the inner arc side and the compressive stress appears on slab surface at the outer arc side at stages 6-8 in straightening zone during continuous casting.Longitudinal cracks generally appear on slab top surface and do not appear on slab bottom surface,which are also observed in industry.展开更多
The effect of deficiency in tunnel crown thickness on the Yellow River Crossing Tunnel with post-tensioned concrete inner lining was investigated by the elasto-plastic finite element method. Changes in the deformation...The effect of deficiency in tunnel crown thickness on the Yellow River Crossing Tunnel with post-tensioned concrete inner lining was investigated by the elasto-plastic finite element method. Changes in the deformations and circumferential stresses of the post-tensioned concrete inner lining with the gradual decrease of the tunnel crown thickness were compared, and the potential bearing risk of insufficient tunnel crown thickness for the Yellow River Crossing Tunnel was revealed. Based on the finite element calculation results of circumferential stresses under different defective cases, the corresponding reinforcement schemes were proposed. The calculation results show that the inner lining can still maintain a satisfactory stress state when the tunnel crown thickness is equal to or greater than 0. 28 m. When the tunnel crown thickness decreases below 0.28 m, the external surface of the crown and internal surface of the crown's adjacent areas may be under tension. The tension stresses will incrementally increase and ultimately exceed the tensile strength of the inner lining concrete as the tunnel crown thickness further decreases gradually. Then, the Yellow River Crossing Tunnel cannot operate normally, and severe cracking, leaking or even failure may occur. When the tunnel crown thickness is equal to or greater than 0.28 m, the reinforcement suggestions are that the void spaces between the inner lining and the outer lining should be back-filled with concrete. When the tunnel crown thickness is less than 0. 28 m, the inner lining should be reinforced by steel plates after concrete back-filling.展开更多
By making use of the direct integration method,an exact analysis of the general three-dimensional thermoelasticity problem is performed for the case of a transversely isotropic homogeneous half-space subject to local ...By making use of the direct integration method,an exact analysis of the general three-dimensional thermoelasticity problem is performed for the case of a transversely isotropic homogeneous half-space subject to local thermal and force loadings.The material plane of isotropy is assumed to be parallel to the limiting surface of the halfspace.By reducing the original thermoelasticity equations to the governing ones for individual stress-tensor components,the effect of material anisotropy in the stress field is analyzed with regard to the feasibility requirement,i.e.,the finiteness of the stress field at a distance from the disturbed area.As a result,the solution is constructed in the form of explicit analytical dependencies on the force and thermal loadings for various kinds of transversely isotropic materials and agrees with the basic principles of the continua mechanics.The solution can be efficiently used as a benchmark one for the direct computation of temperature and thermal stresses in transversely isotropic semi-infinite domains,as well as for the verification of solutions constructed by different means.展开更多
Applying stiffness migration method,a 3D finite element mechanical model is established to simulate the excavation and advance processes.By using 3D nonlinear finite element method,the tunnel boring machine(TBM) excav...Applying stiffness migration method,a 3D finite element mechanical model is established to simulate the excavation and advance processes.By using 3D nonlinear finite element method,the tunnel boring machine(TBM) excavation process is dynamically simulated to analyze the stress and strain field status of surrounding rock and segment.The maximum tensile stress of segment ring caused by tunnel construction mainly lies in arch bottom and presents zonal distribution.The stress increases slightly and limitedly in the course of excavation.The maximum and minimum displacements of segment,manifesting as zonal distribution,distribute in arch bottom and vault respectively.The displacements slightly increase with the advance of TBM and gradually tend to stability.展开更多
The aim of this work is to analyze the stress distributions on a crown-luting cement-substrate system with a finite-element method in order to predict the likelihood of interfacial micro cracks, radial or circumferent...The aim of this work is to analyze the stress distributions on a crown-luting cement-substrate system with a finite-element method in order to predict the likelihood of interfacial micro cracks, radial or circumferential cracks, delamination, fracture and delamination with torsion. The contact and layer interface stresses in elastic layered half-space indented by an elastic sphere were examined using finite element method. The model consists of crown, luting cement and substrate. The solutions were carried out for three different elastic moduli of luting cement. It was placed between the cement and the substrate as a middle layer and its elastic module was chosen lower than the elastic module of crown and higher than the elastic module of dentin. An axisymmetric finite element mesh was set up for the stress analysis. Stress distributions on the contact surface and the interfaces of crown-luting cement and luting cement-dentin have been investigated for three different values of luting cement by using ANSYS. The effects of the luting cement which has three different elastic moduli on the pressure distribution and the location of interfacial stresses of the multi-layer model have been examined. The mechanism of crack initiation in the interfaces and interracial delamination was also studied quantitatively. For each luting cement, the pressure distribution is similar at the contact zone. Stress discontinuities occur at the perfect bonding interfaces of the crown-luting cement and the substrate-luting cement. The maximum stress jumps are obtained for the highest and the lowest elastic module of the luting cement. In the crown-luting cement-substrate system, failures may initiate at crown-luting cement region for luting cement with the lowest elastic module value. In addition, failures at luting cement-substrate region may occur for luting cement with the highest elastic module. In the luting cement, the medium elastic module value is more suitable for stress distribution in crown-luting cement-substrate interfaces.展开更多
Although commonly used, no design method is available for steel web tapered tee section cantilevers. This paper investigates the bending stresses of such beams. Relationships between the maximum compressive stress and...Although commonly used, no design method is available for steel web tapered tee section cantilevers. This paper investigates the bending stresses of such beams. Relationships between the maximum compressive stress and the degree of taper were investigated. An analytical model is presented to determine the location of the maximum stress when subjected to a uniformly distributed load or a point load at the free end and was validated using finite element analysis and physical tests. It was found that the maximum stress always occurs at the support when subjected to a uniformly distributed load. When subjected to a point load at the free end and the degree of taper is up to seven, it was found that Miller's equation could be used to determine the location of the maximum stress. However, it is shown that when the degree of taper is greater than seven, Miller's equation does not accurately predict the location and the analytical model should be used. It was also found that the location of the maximum stress was solely dependent on the degree of taper, while a geometric ratio, fl was required to determine the magnitude of the maximum stress. A simple method that predicts the magnitude of the maximum stress is proposed. The average error in the prediction of the magnitude of the maximum stress is found to be less than 1.0%.展开更多
Combined with a proposed homogeneous earth dam in deep cladding foundation, Duncan E-B model is applied to simulate dam-filled material, apply to three-dimensional nonlinear finite element method, attain the stress-st...Combined with a proposed homogeneous earth dam in deep cladding foundation, Duncan E-B model is applied to simulate dam-filled material, apply to three-dimensional nonlinear finite element method, attain the stress-strain distribution and alteration in concrete cutoff wall in completion and water storage periods, analysis the stress state in the contact element between concrete cutoff wall and cladding foundation, provide the corresponding measures. The calculation results show that the design of concrete cutoff wall and homogeneous earth dam is reasonable.展开更多
The majority of foot deformities are related to arch collapse or instability,especially the longitudinal arch.Although the relationship between the plantar fascia and arch height has been previously investigated,the s...The majority of foot deformities are related to arch collapse or instability,especially the longitudinal arch.Although the relationship between the plantar fascia and arch height has been previously investigated,the stress distribution remains unclear.The aim of this study was to explore the role of the plantar ligaments in foot arch biomechanics.We constructed a geometrical detailed three-dimensional (3-D) finite element (FE) model of the human foot and ankle from computer tomography images.The model comprised the majority of joints in the foot as well as bone segments,major ligaments,and plantar soft tissue.Release of the plantar fascia and other ligaments was simulated to evaluate the corresponding biomechanical effects on load distribution of the bony and ligamentous structures.These intrinsic ligaments of the foot arch were sectioned to simulate different pathologic situations of injury to the plantar ligaments,and to explore bone segment displacement and stress distribution.The validity of the 3-D FE model was verified by comparing results with experimentally measured data via the displacement and von Mise stress of each bone segment.Plantar fascia release decreased arch height,but did not cause total collapse of the foot arch.The longitudinal foot arch was lost when all the four major plantar ligaments were sectioned simultaneously.Plantar fascia release was compromised by increased strain applied to the plantar ligaments and intensified stress in the midfoot and metatarsal bones.Load redistribution among the centralized metatarsal bones and focal stress relief at the calcaneal insertion were predicted.The 3-D FE model indicated that plantar fascia release may provide relief of focal stress and associated heel pain.However,these operative procedures may pose a risk to arch stability and clinically may produce dorsolateral midfoot pain.The initial strategy for treating plantar fasciitis should be non-operative.展开更多
Fiber-metal laminates(FMLs)consist of three layers of aluminum alloy 2024-T3 and two layers of glass/epoxy prepreg,and it(it means FMLs)is laminated by Al alloy and fiber alternatively.Fatigue crack growth rates in no...Fiber-metal laminates(FMLs)consist of three layers of aluminum alloy 2024-T3 and two layers of glass/epoxy prepreg,and it(it means FMLs)is laminated by Al alloy and fiber alternatively.Fatigue crack growth rates in notched fiber-metal laminates under constant amplitude fatigue loading were studied experimentally and numerically and were compared with them in monolithic 2024-T3 Al alloy plates.It is shown that the fatigue life of FMLs is about 17 times longer than monolithic 2024-T3 Al alloy plate;and crack growth rates in FMLs panels remain constant mostly even when the crack is long,unlike in the monolithic 2024-T3 Al alloy plates.The formula to calculate bridge stress profiles of FMLs was derived based on the fracture theory.A program by Matlab was developed to calculate the distribution of bridge stress in FMLs,and then fatigue growth lives were obtained.Finite element models of FMLs were built and meshed finely to analyze the stress distributions.Both results were compared with the experimental results.They agree well with each other.展开更多
The objective of this paper is to investigate the position of the resultant force in involute spline coupling teeth due to the contact pressure distribution for both ideal and misaligned conditions. In general, spline...The objective of this paper is to investigate the position of the resultant force in involute spline coupling teeth due to the contact pressure distribution for both ideal and misaligned conditions. In general, spline coupling teeth are in contact all along the involute profile and the load is far from uniform along the contact line. Theoretical models available in publications consider the resultant contact force as it is applied at the pitch diameter, and this study aims to evaluate the error introduced within the confines of a common approximation environment. This analysis is carried out through using finite element method (FEM) models, considering spline couplings in both ideal and misaligned conditions. Results show that the differences between the load application diameter and pitch diameter are not very obvious in both ideal and misaligned conditions; however, this ap- proximation becomes more important for the calculation of the tooth stiffness.展开更多
基金Projects(51071122,51271147,51201134)supported by the National Natural Science Foundation of ChinaProject(3102014JCQ01023)supported by the Fundamental Research Funds for the Central UniversitiesProject(115-QP-2014)supported by the Research Fund of the State Key Laboratory of Solidification Processing in Northwestern Polytechnical University,China
文摘A three-dimensional cyclic symmetry finite element model of titanium-matrix composites(TMCs) ring was developed to investigate the stress distribution and burst failure. The effects of fiber volume fractions, reinforced areas, thermal residual stresses and two different temperatures on stress distribution were studied. The burst speed was obtained through analyzing the hoop tensile stresses under a series of rotating speeds. The results indicate that at the two different temperatures, the influences of fiber volume fractions and reinforced areas on stress level and distribution are different. Some proposals are provided for the structure design of the TMCs ring. With regard to thermal residual stresses, a larger reinforced area is an advisable choice for design of the ring at higher temperature.
文摘This paper presents finite element modeling (FEM) to simulate the present-day stress field and crustal deformation using NE-SW structural section in the central Seismic Gap region of the Garhwal Himalaya. Our study deals with the effect of geometrical characteristics and rock layer parameters on the upper crust. Modeling results show that two types of tectonic regimes developed in the central Seismic Gap region: the geotectonics of the northern part has been controlled by regional compression, whereas southern part is characterized by regional extension. Correspondingly, thrust faults are induced in the northern part and normal faults are extensively developed in the southern front. Those evidences noticeably indicate that the compressive tectonic environment of the Himalaya becomes change into the extensional tectonic regime in its front. The computed shear stress accumulation along the northern fiat of Main Himalayan Thrust (MHT) implies that considerable amount of interseismic stress is building up along the MHT system in the Himalaya, which ultimately release through the possible future great Himalayan earthquake (M 〉 8). The comparison between our modeled stress field, faulting pattern and horizontal shortening rate with the distribution of the microseismic events, focal mechanism solutions, active faulting and GPS data in the central Seismic Gap region shows good agreement.
基金Project(50634030) supported by the National Natural Science Foundation of ChinaProject(20090042120005) supported by the Doctorate Foundation of the Ministry of Education of ChinaProject(2006CB605208-1) supported by the State Basic Research Program of China
文摘Deformation behavior of slab at the straightening stage during continuous casting was simulated by the explicit dynamic finite element method,and the stress distribution along the width direction of the slab and its change regularity at slab center during continuous casting were obtained.The influence of distribution and change of stress on the propagation of longitudinal cracks on slab surface was discussed.The results show that the tensional stress appears on slab surface at the inner arc side and the compressive stress appears on slab surface at the outer arc side at stages 6-8 in straightening zone during continuous casting.Longitudinal cracks generally appear on slab top surface and do not appear on slab bottom surface,which are also observed in industry.
基金The Natural Science Foundation of Hubei Province(No.2017CFB667)the National Natural Science Foundation of China(No.51079107)
文摘The effect of deficiency in tunnel crown thickness on the Yellow River Crossing Tunnel with post-tensioned concrete inner lining was investigated by the elasto-plastic finite element method. Changes in the deformations and circumferential stresses of the post-tensioned concrete inner lining with the gradual decrease of the tunnel crown thickness were compared, and the potential bearing risk of insufficient tunnel crown thickness for the Yellow River Crossing Tunnel was revealed. Based on the finite element calculation results of circumferential stresses under different defective cases, the corresponding reinforcement schemes were proposed. The calculation results show that the inner lining can still maintain a satisfactory stress state when the tunnel crown thickness is equal to or greater than 0. 28 m. When the tunnel crown thickness decreases below 0.28 m, the external surface of the crown and internal surface of the crown's adjacent areas may be under tension. The tension stresses will incrementally increase and ultimately exceed the tensile strength of the inner lining concrete as the tunnel crown thickness further decreases gradually. Then, the Yellow River Crossing Tunnel cannot operate normally, and severe cracking, leaking or even failure may occur. When the tunnel crown thickness is equal to or greater than 0.28 m, the reinforcement suggestions are that the void spaces between the inner lining and the outer lining should be back-filled with concrete. When the tunnel crown thickness is less than 0. 28 m, the inner lining should be reinforced by steel plates after concrete back-filling.
基金supported by Joint Fund of Advanced Aerospace Manufacturing Technology Research(No. U1937601)the partial financial support of this research by the budget program of Ukraine“Support for the Development of Priority Research Areas”(No.CPCEC 6451230)。
文摘By making use of the direct integration method,an exact analysis of the general three-dimensional thermoelasticity problem is performed for the case of a transversely isotropic homogeneous half-space subject to local thermal and force loadings.The material plane of isotropy is assumed to be parallel to the limiting surface of the halfspace.By reducing the original thermoelasticity equations to the governing ones for individual stress-tensor components,the effect of material anisotropy in the stress field is analyzed with regard to the feasibility requirement,i.e.,the finiteness of the stress field at a distance from the disturbed area.As a result,the solution is constructed in the form of explicit analytical dependencies on the force and thermal loadings for various kinds of transversely isotropic materials and agrees with the basic principles of the continua mechanics.The solution can be efficiently used as a benchmark one for the direct computation of temperature and thermal stresses in transversely isotropic semi-infinite domains,as well as for the verification of solutions constructed by different means.
基金Supported by National Natural Science Foundation of China(No.90815019)National Key Basic Research Program of China("973" Program,No.2007CB714101)Key Project in the National Science and Technology Pillar Program during the Eleventh Five-Year Plan Period(No.2006BAB04A13)
文摘Applying stiffness migration method,a 3D finite element mechanical model is established to simulate the excavation and advance processes.By using 3D nonlinear finite element method,the tunnel boring machine(TBM) excavation process is dynamically simulated to analyze the stress and strain field status of surrounding rock and segment.The maximum tensile stress of segment ring caused by tunnel construction mainly lies in arch bottom and presents zonal distribution.The stress increases slightly and limitedly in the course of excavation.The maximum and minimum displacements of segment,manifesting as zonal distribution,distribute in arch bottom and vault respectively.The displacements slightly increase with the advance of TBM and gradually tend to stability.
文摘The aim of this work is to analyze the stress distributions on a crown-luting cement-substrate system with a finite-element method in order to predict the likelihood of interfacial micro cracks, radial or circumferential cracks, delamination, fracture and delamination with torsion. The contact and layer interface stresses in elastic layered half-space indented by an elastic sphere were examined using finite element method. The model consists of crown, luting cement and substrate. The solutions were carried out for three different elastic moduli of luting cement. It was placed between the cement and the substrate as a middle layer and its elastic module was chosen lower than the elastic module of crown and higher than the elastic module of dentin. An axisymmetric finite element mesh was set up for the stress analysis. Stress distributions on the contact surface and the interfaces of crown-luting cement and luting cement-dentin have been investigated for three different values of luting cement by using ANSYS. The effects of the luting cement which has three different elastic moduli on the pressure distribution and the location of interfacial stresses of the multi-layer model have been examined. The mechanism of crack initiation in the interfaces and interracial delamination was also studied quantitatively. For each luting cement, the pressure distribution is similar at the contact zone. Stress discontinuities occur at the perfect bonding interfaces of the crown-luting cement and the substrate-luting cement. The maximum stress jumps are obtained for the highest and the lowest elastic module of the luting cement. In the crown-luting cement-substrate system, failures may initiate at crown-luting cement region for luting cement with the lowest elastic module value. In addition, failures at luting cement-substrate region may occur for luting cement with the highest elastic module. In the luting cement, the medium elastic module value is more suitable for stress distribution in crown-luting cement-substrate interfaces.
文摘Although commonly used, no design method is available for steel web tapered tee section cantilevers. This paper investigates the bending stresses of such beams. Relationships between the maximum compressive stress and the degree of taper were investigated. An analytical model is presented to determine the location of the maximum stress when subjected to a uniformly distributed load or a point load at the free end and was validated using finite element analysis and physical tests. It was found that the maximum stress always occurs at the support when subjected to a uniformly distributed load. When subjected to a point load at the free end and the degree of taper is up to seven, it was found that Miller's equation could be used to determine the location of the maximum stress. However, it is shown that when the degree of taper is greater than seven, Miller's equation does not accurately predict the location and the analytical model should be used. It was also found that the location of the maximum stress was solely dependent on the degree of taper, while a geometric ratio, fl was required to determine the magnitude of the maximum stress. A simple method that predicts the magnitude of the maximum stress is proposed. The average error in the prediction of the magnitude of the maximum stress is found to be less than 1.0%.
文摘Combined with a proposed homogeneous earth dam in deep cladding foundation, Duncan E-B model is applied to simulate dam-filled material, apply to three-dimensional nonlinear finite element method, attain the stress-strain distribution and alteration in concrete cutoff wall in completion and water storage periods, analysis the stress state in the contact element between concrete cutoff wall and cladding foundation, provide the corresponding measures. The calculation results show that the design of concrete cutoff wall and homogeneous earth dam is reasonable.
基金supported by the National Natural Science Foundation of China(Grant No. 30801163)
文摘The majority of foot deformities are related to arch collapse or instability,especially the longitudinal arch.Although the relationship between the plantar fascia and arch height has been previously investigated,the stress distribution remains unclear.The aim of this study was to explore the role of the plantar ligaments in foot arch biomechanics.We constructed a geometrical detailed three-dimensional (3-D) finite element (FE) model of the human foot and ankle from computer tomography images.The model comprised the majority of joints in the foot as well as bone segments,major ligaments,and plantar soft tissue.Release of the plantar fascia and other ligaments was simulated to evaluate the corresponding biomechanical effects on load distribution of the bony and ligamentous structures.These intrinsic ligaments of the foot arch were sectioned to simulate different pathologic situations of injury to the plantar ligaments,and to explore bone segment displacement and stress distribution.The validity of the 3-D FE model was verified by comparing results with experimentally measured data via the displacement and von Mise stress of each bone segment.Plantar fascia release decreased arch height,but did not cause total collapse of the foot arch.The longitudinal foot arch was lost when all the four major plantar ligaments were sectioned simultaneously.Plantar fascia release was compromised by increased strain applied to the plantar ligaments and intensified stress in the midfoot and metatarsal bones.Load redistribution among the centralized metatarsal bones and focal stress relief at the calcaneal insertion were predicted.The 3-D FE model indicated that plantar fascia release may provide relief of focal stress and associated heel pain.However,these operative procedures may pose a risk to arch stability and clinically may produce dorsolateral midfoot pain.The initial strategy for treating plantar fasciitis should be non-operative.
基金supported by Chengdu Aircraft Design&Research Institutethe National Natural Science Foundation of China(Grant No.11002111)the Scholarship of Study Abroad Council
文摘Fiber-metal laminates(FMLs)consist of three layers of aluminum alloy 2024-T3 and two layers of glass/epoxy prepreg,and it(it means FMLs)is laminated by Al alloy and fiber alternatively.Fatigue crack growth rates in notched fiber-metal laminates under constant amplitude fatigue loading were studied experimentally and numerically and were compared with them in monolithic 2024-T3 Al alloy plates.It is shown that the fatigue life of FMLs is about 17 times longer than monolithic 2024-T3 Al alloy plate;and crack growth rates in FMLs panels remain constant mostly even when the crack is long,unlike in the monolithic 2024-T3 Al alloy plates.The formula to calculate bridge stress profiles of FMLs was derived based on the fracture theory.A program by Matlab was developed to calculate the distribution of bridge stress in FMLs,and then fatigue growth lives were obtained.Finite element models of FMLs were built and meshed finely to analyze the stress distributions.Both results were compared with the experimental results.They agree well with each other.
文摘The objective of this paper is to investigate the position of the resultant force in involute spline coupling teeth due to the contact pressure distribution for both ideal and misaligned conditions. In general, spline coupling teeth are in contact all along the involute profile and the load is far from uniform along the contact line. Theoretical models available in publications consider the resultant contact force as it is applied at the pitch diameter, and this study aims to evaluate the error introduced within the confines of a common approximation environment. This analysis is carried out through using finite element method (FEM) models, considering spline couplings in both ideal and misaligned conditions. Results show that the differences between the load application diameter and pitch diameter are not very obvious in both ideal and misaligned conditions; however, this ap- proximation becomes more important for the calculation of the tooth stiffness.