This study determines the plastic, shakedown and elastic limit loads and the corresponding regimes of 90° pressurized pipe bends with shape imperfections subjected to cyclic in-plane closing bending ...moment. Three dimensional FE analyses with small strain formulation was performed. The induced shape imperfections are ovality and thinning, which were varied from 0% to 20% with a 5% increment at every step. Plastic limit load was determined from the moment rotation curve while shakedown and elastic limit loads were determined by using Abdalla's simplified technique (ST). The analysis indicates that the presence of ovality produces considerable effect on plastic limit loads. Thinning alters plastic limit loads only for long bend radius models with no ovality when normalised internal pressure ratio is unity and in other cases its effect is minimal. Presence of ovality and thinning has a remarkable effect on shakedown and elastic limit loads and their regimes though the thinning effect is insignificant at higher ovality (10%–20%). Bree diagram was constructed for the reference pipe bend models. It was observed that the failure of pipe bends is due to either reversed plasticity or ratcheting. Mathematical equations are proposed to express the generated plastic, shakedown and elastic limit moment boundaries for the analyzed pipe bend models.
•Plastic, shakedown and elastic limit loads for shape imperfect 90ᴼ pipe bends subjected to internal pressure and cyclic in-plane closing bending moment was studied.•Shape imperfections in the pipe bends considered for the analysis are ovality and wall thinning.•Moment rotation curve is used for determining plastic limit load and Abdalla's simplified technique is employed to determine the shakedown and elastic limit loads.•Premature failure cause of pipe bends either due to reversed plasticity or ratcheting is studied by using bree diagram.•Beyond a certain pressure limit, ovality increases the pipe bend’s performance.
This paper develops rigorous lower-bound shakedown solutions for pavement structures subjected to Hertz loads both in plane strain and three-dimensional (3D) models. Firstly, a necessary shakedown ...condition is derived based on Melan's shakedown theorem and the use of Prandtl-Reuss theory in conjunction with von Mises criterion, which gives a maximum value to the rigorous shakedown limit. Secondly, the rigorous lower-bound shakedown solution is established uses a self-equilibrated critical residual stress field with an optimisation procedure. As expected, the necessary shakedown condition gives larger values of normalised shakedown limits than that of the normalised rigorous lower-bound shakedown solution, with the maximum difference of 93%. Also observed is that the 3D model always overestimates the normalised rigorous shakedown limits of the plane strain model by approximately 8-13%, while the two models capture approximately the same trend of normalised rigorous shakedown limits varied with frictional coefficient. Finally, the parametric studies on normalised rigorous shakedown limit of pavements in terms of frictional coefficient and Poisson's ratio are performed. It shows that the normalised rigorous shakedown limits are proportional to material yield stress but decrease markedly with raising frictional coefficient; while the effect of the Poisson's ratio on normalised rigorous shakedown limits is marginally. In addition, the critical residual stress fields lie between two residual stress limits, which intersect beneath the surface for small μ, while tend to converge at the half-space surface when μ = 0.3, indicating that the failure mode of the pavement changes from subsurface failure to surface failure when increasing the frictional coefficient μ. As a result, the present shakedown solutions can provide reference to the design of pavements subjected to rolling and sliding loading contact.
•Revealing shakedown mechanisms of four-dimensional Bree diagram under three loading conditions for the first time.•Constructing shakedown boundaries of four-stress parameters Bree-type ...problems.•Proposing a unified and conservative shakedown assessment scheme.•Discussion of mechanical models for various uniaxial Bree-type problems.•Developing a common finite element model for all discussed Bree-type problems.
For pressure equipment in the industrial field, the shakedown design methods presented in the ASME and other international design codes are mainly based on the Bree problem. The original Bree problem and its related modified Bree problems usually contain two types of stress parameters, whereas design by analysis based on the stress categorisation of the codes actually includes multiple types of stress parameters. In this paper, the four-stress parameters Bree-type shakedown problems (involving mechanical membrane and bending as well as thermal membrane and bending) have been studied under three typical thermo-mechanical loading conditions for the first time to reveal the mechanisms of the four-dimensional Bree diagram. A detailed discussion on the mechanical models for various uniaxial Bree-type problems is conducted to clarify the scope of application and mechanical assumptions for each problem. A common finite element model is developed for all discussed Bree-type problems. The Linear Matching Method (LMM) is utilised to construct the shakedown boundaries of three examples under three loading conditions, and the semi-analytical parametric equations of the shakedown boundaries are derived based on the numerical results and four-dimensional ratcheting boundary theory. Using the idea of a minimum shakedown boundary, a unified and conservative shakedown assessment scheme is proposed for the four-stress parameters Bree-type problems. The newly presented results can enhance the understanding of the structural behaviour of pressure equipment for many loading conditions and guide shakedown design and ratcheting assessment of practical industrial components.
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•A multi-layered model of functionally graded coating-substrate structure is proposed.•The contact elastic shakedown map of functionally graded structure is obtained.•The factors affecting the ...contact shakedown are discussed.
The elastic shakedown analysis of rolling/sliding contact with friction heat is essential for the safety evaluation of tribological components with a functionally graded (FG) coating. However, the corresponding works are still lacking due to the complexity of thermo-elastic contact stress calculation and residual stress optimization induced by the non-homogeneity of material properties. An effective multi-layered model for the thermo-elastic rolling/sliding contact of two similar elastic cylinders with the FG coating involving arbitrarily distributed thermo-elastic properties is established, and the corresponding thermo-elastic stress field is obtained subsequently by using the conjugate gradient method and fast Fourier transform algorithm. Then, a specific bisection iteration procedure is applied to evaluate the elastic shakedown limit (ESL), and the effects of friction coefficient, rolling speed, distribution gradient indexes of various thermo-mechanical properties, and FG coating thickness on the ESL are discussed, respectively. The results show that the increases of friction coefficient and rolling speed can cause the decrease of ESL, the ESL can be improved significantly by adjusting the distribution gradients of shear modulus and yield stress appropriately, and the relations between the ESL and various heat conduction parameters show a strong dependence on the sliding velocity.
This study provides a clear insight into the dynamic response of rectangular plates subjected to repeated impacts using experimental and numerical methods. Repeated impacts are performed on the mild ...steel plates with a length-to-width ratio of 3, which can be regarded as scaled-down ship plates. During the experiments, both fracture and pseudo-shakedown phenomena are observed, occurring under relatively high and low impact velocities, respectively. By thoroughly analysing various parameters related to the dynamic response, including the plate deformation, contact force, rebound velocity of the impactor, and dissipated energy ratio, among others, the study effectively characterizes the behaviour of plates under repeated impacts. Numerical simulations are conducted and a detailed comparison between the numerical and experimental results is carried out, accompanied by a careful analysis of the discrepancies between them. The numerical simulations exhibit a qualitative capability to predict the dynamic response of plates subjected to repeated impacts. The ‘failure displacement’ proposed by the authors is verified against the experimental results. Moreover, the threshold impact velocity for the pseudo-shakedown occurrence is unveiled by numerical simulations.
•Repeated impacts are performed on the mild steel plates with a length-to-width ratio of 3.•Fracture and pseudo-shakedown are observed in the experiments, under relatively high and low impact velocities.•Numerical simulations are conducted and a detailed comparison with the experimental results shows good results.•The numerical analyses identify the threshold impact velocity for the pseudo-shakedown occurrence.
Elastoplastic design of process equipment in petrochemical and power generation industries requires the consideration of thermal ratcheting. Classical Bree diagrams assuming no strain hardening have ...been the basis in most of today's design Codes and Standards, which can be excessively conservative. In this study, three major kinematic hardening models by Pager, Armstrong and Frederick, and Chaboche are considered for modeling detailed thermal ratcheting behaviors and establishing Bree diagrams. It is found that the linear hardening model by Prager provides a reasonable upper bound ratcheting strain estimation while the perfect plastic assumption by Bree is adequate for the design diagram construction.
•Elastoplastic design based on Bree diagram is revisited considering kinematic strain hardening.•5 different hardening model are compared on their effect on Bree diagram construction.•The Prager hardening model balances the simplicity and accuracy in Bree diagram construction.
•Cyclic hardening and cyclic saturation appear in the plastic shakedown state.•Investigating deformation mechanisms during plastic shakedown and ratcheting.•AE signals monitor the fatigue process in ...plastic shakedown and ratcheting.
Fatigue performance and durability application have always been the key issues impeding the understanding of aero-engine pressure pipe due to its non-standardized shape, making tests more challenging. This paper constructs an optimized testing system to elucidate the plastic shakedown and ratcheting behavior of 1Cr18Ni10Ti pressure pipe subjected to in-service loadings. Moreover, microstructural evolution's role in regulating the cyclic plasticity behavior is qualitatively established. Experimental results indicate that the pipe deformation exhibits compressive ratcheting behavior in plastic shakedown state, and the initial cyclic hardening followed by cyclic saturation occurs during the whole cyclic response. This corresponds to a decreasing ratcheting displacement rate followed by a steady state appears. Except for the first two similar deformation stages mentioned above, the abrupt cyclic softening and the resultant increasing ratcheting displacement rate are obtained at the final fracture stage of the pipe deformation in ratcheting state. A series of detailed microstructural characterizations, including optical microscopy (OM), scanning electron microscopy (SEM), electron backscattered diffraction (EBSD), transmission electron microscopy (TEM), and acoustic emission (AE) are combined to reveal the underlying microstructural mechanisms during cyclic deformation. The volume fraction of austenite and ferrite and the average grain size change during different stages of plastic shakedown and ratcheting deformation. The synergistic effects of dislocation structure multiplication, destruction, and dissolution at different stages of plastic shakedown and ratcheting deformation dominate the cyclic plasticity behavior.
•GDS dynamic triaxial tests are presented for shakedown analysis of subgrade material.•Patterns of accumulated axial plastic strains for subgrade material are analyzed.•Critical dynamic stresses for ...plastic shakedown and plastic creep state are obtained.•An empirical model for critical dynamic stresses of subgrade material is proposed.
Coarse-grained soil is widely used as a filling material in the railway subgrade to provide foundational support to the long-term effects of dynamic (cyclic) train loads transmitted by the track structure. Despite this wide usage, there is limited literature on its permanent deformation behavior characterization under different magnitudes of dynamic stress. The GDS (Geotechnical Digital System) dynamic triaxial tests were presented in this paper to characterize and quantify the dynamic response-behavior of coarse-grained railway subgrade soils when subjected to dynamic loading. Based on the shakedown concept, the patterns of accumulated axial plastic strains for the coarse-grained soil under different magnitudes of dynamic stress were classified into three categories, namely plastic shakedown, plastic creep, and incremental collapse. As a result, the critical dynamic stresses for plastic shakedown and plastic creep state could easily be determined. The experimental test results indicated that whilst the critical dynamic stress of the coarse-grained soil increased with an increase in the effective confining pressure, it correspondingly decreased as the loading frequency increased from 1.0 to 3.6 Hz, respectively. From the regression fitting-analysis of the experimental test results, an empirical model is proposed in this paper, which denotes the critical dynamic stress of coarse-grained subgrade soils as a function of the effective confining pressure. As a contribution to the literature and potential engineering applications, the proposed models may provide theoretical significance for the subgrade state evaluation and settlement control.
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