Limit Bending Moment Research Articles

Damage constitutive model of rock and application based on strength theory

Abstract Extending the one-dimensional damage constitutive model of rock materials to 3 dimensions using classical methods fails to capture the significant differences in tensile, compressive, and shear strengths exhibited by the rock materials. Consequently, it is necessary to revise the existing damage constitutive model to describe the damage evolution law and constitutive relationship of rock materials more accurately and provide a theoretical basis for the safety and stability analysis of underground engineering more scientifically, thus ensuring the sustainable development of underground engineering. By introducing the Weibull distribution function and building upon strength theory, a correction function was established. This correction function adjusted the equivalent strain, enabling the development of a 3-dimensional damage constitutive model that accounted for the varying tensile, compressive, and shear strengths of rock materials. The impact of various parameters on the model's fitting effectiveness was evaluated, and a comparative analysis was conducted against pertinent experimental results. Using the theory of neutral axis deviation, the relationship between bending moment and damage variables in a purely bending rock beam was derived. The study revealed that all parameters of the damage constitutive model could be derived from the uniaxial stress-strain curve, and its theoretical findings exhibited strong agreement with experimental results obtained from rock and rock-like materials under uniaxial tension, compression and triaxial compression. Based on the examined cases, it was concluded that, when considering both tensile and compressive damage, the ultimate bending moment of a rock beam was approximately one-third of its elastic limit bending moment in an undamaged state. The results have verified the feasibility of the damage constitutive model.

Lateral impact resistance of full-scale concrete-filled weathering steel tube columns

This paper conducted lateral impact tests on 10 concrete-filled weathering steel tubular (CFWST) columns and studied the effect of prestressing on their resistance to lateral impact. The results indicate that by using prestressed cantilever members, the impact resistance of the column can be significantly improved. This is mainly due to the reduction of tensile stress level in the steel tube at the fixed end of the cantilever column and of the overall plastic deformation of the member, which is achieved through the application of prestress. Furthermore, the test process was also simulated numerically. Based on verifying the correctness of the numerical simulation, the parameters of the full-scale model are analyzed. The impact resistance of the model was evaluated by considering various factors such as impact angle, impact point height ratio, prestress ratio, width-thickness ratio, slenderness ratio, and mass ratio. Based on these analyses, empirical formulas for the lateral ultimate displacement and bending moment of the cantilever members were derived. Particularly, considering the difference in yield strength between tension and compression sides, the theoretical calculation of plastic limit bending moment of concrete-filled square steel tubular members is modified. Additionally, the damage level of the component is defined, and the damage limit value of the concrete-filled steel tubular cantilever column is given based on the different damage grades. On this basis, considering the bending bearing capacity and displacement response of the component, an impact resistance design approach based on the performance of the component is proposed.

Theoretical study on the propagation of high impact energy in the rotor with local plastic deformation after blade off

A theoretical model of an elastic supported rotor system under impact by blade off is established based on d'Alembert principle. The gyroscopic moment of the blade-disk with asymmetrical inertia is considered in the dynamic model. The formation criterion of the plastic hinge in the cylinder rotor are derived and applied in the dynamic model to consider the local plastic deformation of the rotor with large deformation. This model extends the impact model of non-rotating beam with rigid support to the rotating beam and can describe the main mechanical properties of the rotor with blade off. The influence of the main factors, such as sudden unbalance, gyroscopic moment, geometric dimensions and concentrated mass, on the propagation of the impact energy in the rotor are analyzed. The gyroscopic moment can suppress the wave propagation characteristics of the impact energy in a rotating beam, resulting in an overall bending deformation, compared with the wave propagation characteristics of the impact energy in the non-rotating beam. Larger local concentrated mass can reduce the effect of impact, and increasing the cross-section size can significantly improve the plastic limit bending moment of the rotor. The variation of the bending moment in the low-pressure rotor is mainly determined by gyroscopic moment and bending deformation. Due to the constraint of the 1# and 2# pivots, the strain energy generated by fan blade off is mainly distributed in the shaft segment in front of the 2# pivot. Therefore, the impact resistance of this shaft segment should be improved to fulfill the safety design requirement of fan blade off failure.

Iterative Optimal Design of Special Moment Resisting Devices for Steel Frames

The present paper proposes an iterative procedure devoted to reaching the optimal design of an innovative, recently proposed, moment resisting device. This special device, called Limited Resistance Plastic Device (LRPD), can be utilized, as an example, to equip a steel frame when it is required that the frame must be designed to substitute a masonry panel, i.e., it must be characterized by a structural behaviour as close as possible to the one of the replaced masonry wall. This purpose can be reached by designing the relevant frame imposing appropriate constraints on the elastic stiffness and on the limit resistance. The result can be obtained just by ensuring that the elastic stiffness and the limit resistance be independent of each other. To this aim it is necessary to suitably equip the steel frame by the previously cited LRPD. In particular, these moment resisting connections ensure that in a prefixed portion of the given beam element, the limit bending moment reduces without any variation of the global elastic stiffness. In order to reach this goal, the LRPD is substantially constituted by an inner portion, devoted to exhibit the desired reduced limit bending strength and to receive the plastic deformations, and two outer portions, devoted to guarantee the invariance of the elastic bending stiffness. The proposed iterative procedure allows to design a device respecting all the required features avoiding the presence of any dangerous local instability phenomenon. To this aim, the design will contain appropriate constraints ensuring that the device cross sections appertain to class 1 or class 2 I-cross section, as prescribed in the more recent standard codes. Some examples, validated by 3D solid tetrahedral elements analysis in ABAQUS environment, prove the good reliability of the proposed procedure and show the easy applicability of the computational approach.

Limit load for cylinders with axial internal surface cracks under combined internal pressure and global bending moment

The global and local limit loads for cylinders with axial internal surface cracks under combined internal pressure and global bending moment are investigated using the elastic-perfectly plastic finite element (FE) method. An ellipse function is recommended to describe the relationship between the normalised global limit pressure and normalised global limit bending moment. The predictions using the ellipse function are conservative compared with the global FE results except for the cases with bending moment dominance, where the predictions may be slightly non-conservative. The investigation shows that the local limit load FE results can be well described by the local limit load model recommended by Lei and the predictions are all reasonably conservative except for cases with very long and deep cracks. Further work is required to validate the recommended global and local limit load solutions for J predictions when they are used in the reference stress J scheme.

Analytical investigation on deformation of PELE projectile and opening damage to concrete target

A penetrator with enhanced lateral effects (PELE), which is a particular thin-walled structure, is characterized by a low-density filling material surrounded by a high-density jacket material. When the PELE projectile penetrates a concrete target, the target is destroyed by its thin-walled jacket to a large opening. An analytical model describing deformation mode of the jacket during the penetration process is presented. Due to opening diameter of target plate is larger than projectile diameter under condition of a thin concrete target, a relationship between opening diameter of target plate and projectile diameter, based on existing experimental results of ridge projectile penetrating thin concrete target, is formulated. The relationship combined with the analytical model is used to predict opening diameter of concrete target. A quantitative comparison between the jacket deformation mode obtained from analytical results and experimental results is made to validate the accuracy of the analytical model proposed. As a result, the analytical results exhibit good correlation with experimental observations in terms of the opening diameter of target plate and the deformation mode of projectile jacket. In addition, the influence mechanism of three factors (jacket strength, impact velocity, and target thickness) on the opening diameter is analyzed. The jacket strength affects the opening diameter by changing the plastic limit bending moment of the jacket. Even though the impact velocity affects the opening diameter by changing the interaction force between the jacket and the target, it is equivalent to indirectly changing the jacket strength. However, the target thickness affects the opening diameter by changing the penetration time.

Safety Analysis of Carbon Fiber Composite Subjected to Different Levels of Bending Moment

Carbon fiber composite rods can be widely employed in engineering due to their high ratio of the strength and density. Carbon fiber composite rods can be only used in the suitable conditions. In particular, under the bending deformation, the failure of carbon fiber composite rods can lead to the engineering accident if the applying bending moment is beyond a certain value. In this paper, a finite element model (FEM) for carbon fiber composite rod was put forward to simulate the mechanical responses of carbon fiber composite rod subjected to bending moment loading. A series of simulations under different levels of bending moments were conducted. Based on the calculated results, the limit bending moment was obtained by comparing the mechanical responses and the failure strength.

Study on Limit Load of Orthotropic Cylindrical Pipe Under Different Combined Load Conditions

Abstract In engineering, many pressure pipes are made of steels with good plasticity, which are subject to internal pressure, axial force, shear force, bending moment, torsion moment or their combined loads. The plastic limit load is an important indicator of the load capacity of pressure pipe. According to Hill yield function, the theoretical solutions of limit load of orthotropic cylindrical pipe under various combined loads under internal pressure, axial force, shear force, torsion moment, and bending moment have been derived on the basis of elastic perfectly plastic constitutive model. The effects of radial stress on different combined limit loads of cylindrical pipe are explored and these results show that the radial stress should be considered about the limit load calculation especially for thick-walled cylindrical pipe. The interactions of various load combination are analyzed in detail and drawn with the interaction curves. For isotropic cylindrical pipe, the limit load increases with the yield strength. For the orthotropic cylindrical pipe, the limit loads of cylindrical pipe under axial force, bending moment, shear force, and torsion moment without internal pressure are only related to the axial yield strength. The limit bending moment is mainly dependent on the axial yield strength when internal pressure is lower, while the impact of the circumferential yield strength of orthotropic cylindrical pipe is obvious when internal pressure is some higher. When the axial yield strength of orthotropic cylindrical pipe is the same, the circumferential yield strength can enhance the limit axial load, limit torsion moment, and limit shear load. Under the different load conditions including internal pressure, bending moment, axial force, shear force, and torsion moment or their combined loads, the relation of limit bending moment with yield strength ratio is diverse, which is decide by the load combination, the circumferential yield strength, and the axial yield strength.

Research on static performance of T-shaped stiffener Reinforced Box Space joints under Spatial Loading

The problem of inner partition reinforced box column are the weld quality, time-consuming and laborious. In view of the problem, T-shaped stiffener externally reinforced box space beam-column joinst are proposed in this paper. The finite element simulation was used to study the evolution of the initial stiffness of the T-shaped stiffener reinforced joints under spatial static loading, and the mechanism of the mutual force transfer between the T-shaped stiffener and the H-shaped beam and square column are revealed. The beam section height, the width of the T-shaped stiffener web, the height of the T-shaped stiffener flange are analyzed which are the T-shaped stiffener geometric parameters. The static performance of the reinforced box space joints are gained. The results show that the plastic hinge of the T-shaped stiffener reinforced the space joint is formed in the distance from the beam flange which is the point connecting with the end of the T-shaped stiffener. Such as the stress distribution is reasonable. The bearing capacity, plastic limit bending moment and initial stiffness of the box space joint can be effectively improved with increasing of the beam section height, stiffener web width and stiffener flange height. Therefore, the external T-shaped stiffener reinforced square steel column H-beam space joint has good static performance.

Effect of projectile parameters on opening behavior of PELE penetrating RC target

When a penetrator with enhanced lateral effect (PELE) impacts on a reinforced concrete (RC) target, the target is damaged with a large opening. An understanding of how PELE projectile parameters affect the opening dimension, is essential for effective design of the PELE projectile. In this study, under the condition that the impact velocity and target parameters (strength and thickness) were fixed values, the important influence factors of the PELE (jacket wall thickness B, jacket material strength Y1, filling material strength Y2 and angle of monolithic jacket θ) were determined by a dimensional analysis. Tests and simulations of the PELE penetrating the RC target were conducted to analyze the influence of these factors on opening diameter (D¯, an equivalent diameter under relative kinetic energy). Based on the test and simulation results, it is found that the influence of these factors B, Y1 and θ on the deformation mode of the jacket shows a similar trend: as values of the three factors decrease, the jacket deforms from small bending deformation to large one, and then to curling deformation. This causes the opening diameter to first increase with the decrease of these three factors, and then decreases. It is well known that the bending resistance of the jacket is related to these factors B, Y1 and θ. Therefore, a plastic limit bending moment (M0) of the jacket was quoted to characterize the influence of these factors on the bending deformation of the jacket and the opening diameter of the target. The influence factor Y2 causes D¯ to first increase with the increase of Y2, and then decreases. A formula was developed to predict the opening diameter, whose influence parameters were considered in a dimensionless way. It has been shown that the dimensionless opening diameter D¯/d1 is dependent on two dimensionless parameters Q=d13fcM0 and G=fcY2, where d1 and fc are the outer diameter of the projectile and the compressive strength of the target, respectively.

Limit and Shakedown Analysis of 45-Degree Piping Elbows Under Internal Pressure and In-Plane Bending

Abstract This paper carries out the limit and shakedown analysis of 45 deg piping elbows made up of elastic–perfectly plastic materials by means of the recently proposed stress compensation method (SCM). The elbows are subjected to steady internal pressure and cyclic in-plane closing, opening, and reversed bending moments. Different geometries of the piping elbows and various combinations of these applied loads are investigated to generate various plastic limit and shakedown limit load interaction curves. The plastic limit bending moment and plastic limit internal pressure calculated with the SCM are compared to those determined by the twice-elastic-slope approach. Full step-by-step (SBS) elastic–plastic incremental finite element analysis (FEA) is utilized to verify the structural cyclic responses on both sides of the curves obtained and further to confirm the correct shakedown limit loads and boundaries. It is shown that the SCM calculates the shakedown limit load accurately and possesses about 40 times the computation efficiency of the SBS elastic–plastic incremental method. The effects of the ratios of mean radius to wall thickness and bending radius to mean radius of the piping elbow as well as the loading conditions on the plastic limit and shakedown limit load interaction curves are presented. The results presented in this work give a comprehensive understanding of long-term response behaviors of the piping elbow subjected to cyclic loadings and provide some guidance for the design and integrity assessment of piping systems.

Preparation of FRC/Steel by Lap-joint and Its Mechanical Properties

The Lap-joint configuration and the vacuum assisted forming method were put forward to meet the demands of the compression bending strength and the strong connection between the steel and FRC. The bending mechanic tests of the Lap-joint bending specimens carried out, the fabricate processing stability, limit carrying capacity and failure modes were analyzed. The experimental results show that the fabricate processing of the Lap-joint is stable; the bending stiffness is 19.5 N/mm, and the limit bending moment reaches 318 N∙m. The main failure modes are the failure of core material at junction of sandwich plate and steel, and the bending failure in the position of the inner and outer surface.

Structural Control of a Wind Turbine Accounting for Second Order Effects

The negative impact of the use of fossil fuels on the environment has lead to a boom in the production of wind turbines. The progressively increasing turbines’ height, decided to take ad-vantage of the smoother winds at higher altitude, has led to an increased demand to control tower forces. The proposed work is focused on the application of a semi-active (SA) control system to limit bending moment demand at the base of a wind turbine by relaxing the base restraint of the turbine’s tower, without increasing the top displacement. The proposed SA control system reproduces a variable restraint at the base that changes in real time its mechanical properties according to the instantaneous response of the turbine’s tower. This smart restraint is made of a central smooth hinge, elastic springs and SA magnetorheological dampers driven by a properly designed control algorithm. A commercial 105 m tall wind turbine has been considered as a case study. Several numerical simulations have been performed with reference to two extreme loads, different one each other for intensity, duration, frequency content, so as to understand if a unique optimal configuration of the controller can be defined for both of them. The proposed study is also focused on understanding whether and how to reduce the residual top displacement due to the possible incremental base rotation that may happen during a wind load history, especially when it is long lasting.

Limit bending moment for pipes with two circumferential flaws under combined internal pressure and bending

By assuming the neutral axis is perpendicular to the bending plane, this paper presents a general analytical solution to predict the limit bending moment for pipes with two inner surface circumferential flaws under combined internal pressure and bending moment. The circumferential stress caused by internal pressure is taken into account through Mises yield criterion. Based on the general solution, expressions for symmetrical and unsymmetrical bending with two flaws of the same size are proposed. Then the influence of perpendicularity assumption is discussed and it demonstrates that the deviation caused by perpendicularity assumption is negligible. Finite element analyses and experiments are both carried out. The results calculated by proposed expressions are found to be in good agreement with both numerical results and experimental ones. The effect of several parameters on the limit bending moment is also studied. Results reveal that circumferential stress caused by internal pressure has important influence on bending moment, and should be taken into account.

Limit Load Analysis of Elbow with Local Wall Thinning under Combined Loads

The local wall thinning(LWT) is a kind of common volume defect in pressure pipe. The limit loads of elbows with LWT under pressure, bending moment, torque and their combined loads have been studied in detail by orthogonal experimental design and finite element method. The results have shown that the influence of depth and circumferential length of LWT on the limit load is more obvious compared to that of axial length when an elbow is under pressure, bending moment or torque. The change of limit bending moment and torque with the depth of LWT and circumferential length is significant for an elbow under combined bending moment and torque. At last, the safety assessment equations for elbow under combined in-plane closing bending moment and torque were proposed by regression analysis.

Reverse Bending Research of Leveling on Plastic Hardening Material

In order to further study the relevance between the first and the second reserve bending in the process of parallel roller level of plastic hardening material. Roller level bending mechanical model is established based on hardening elastic-plastic material in accordance with the fundamental of engineering elastic-plastic mechanics. The stress relevance of the two reserve bending processes is described by analytical method and discussed out the changing process of residual stress distribution, moment and loading stress with the section bending process of the roller level. With this model it is demonstrated that moment ratio and curvature ration is not simple "M-C" relation any longer, but function related to the first curvature ratio, the second curvature ratio and hardening factors. It is theoretically proved the original curvature theory can not apply to multi-reserve bending leveling processes.Analytical results show that during the second reserve bending process, there are decreased elastic limit bending moment and the bending moment in leveling process but increased spring-back ratio. Considered hardening of the same material residual stress mean square deviation are 71.2 and 65.6, average rebound ratio are 0.49 and 0.43.

An improved method for optimal shakedown design of circular plates

The paper analyzes the problem of distributing an optimal limit bending moment of elastic-plastic circular and annular plates subjected to variable repeated loading at shakedown. The geometry of the plate is known and variable repeated loading is defined by time-independent upper and lower bounds (unloading phenomenon of a cross section is ignored). Equilibrium finite elements are applied for a discrete model. The safety of the plate is described with reference to nonlinear von Mises yield criterion while serviceability – by displacement limitations. The analysis problem of internal forces and deformations is formulated as a complete system of equations for an elastic-plastic plate in the shakedown state. The implementation of compatibility conditions for residual displacements and MATLAB nonlinear optimization tools allowed creating an improved mathematical model for optimizing the plate and its effective numerical realization. Research methods and numerical results are based on the assumption of small deformations. DOI: http://dx.doi.org/10.5755/j01.mech.20.4.6542

Limit loads for pipe bends under combined pressure and out-of-plane bending moment based on finite element analysis

In present work, limit loads for pipe bends under combined internal pressure and out-of-plane bending moment have been investigated systematically via finite element simulation based on both small displacement analysis and large displacement analysis. By normalized methods, a wide range of non-dimensional parameters were considered and limit bending moment expression has been suggested which shows that r/t is the main factor affecting the limit bending moment. In addition, the effect of internal pressure on the limit loads was studied in detail which is significant for thin walled pipe bends. It is found that yield strain ε0 has an obvious effect on limit load based on large displacement analysis. According to the finite element results, practicable engineering assessment equations of limit loads have been proposed for pipe bends under combined loads. These proposed equations are validated with experiment data and also justified to be a good choice for limit loads assessment.

Testing of Elastic Stresses and Limit Close-Bending Moments of Reducer Bends Subject to In-Plane Close-Bending Moments

Elastic stresses and limit close-bending moments of reducing elbows subject to in–plane close-bending moments were validated by limit element analysis (LMA) and testing. Results of LMA show that if the end bend moment M acts on small end while the large end is fixed, the largest movement acts on neutral line near the larger end, and large circumferential stress lies on inside surface near neutral lines but deflect to intrados of large end. For the same model, if the end bend moment M acts on large end while the small end is fixed, the largest movement acts on near the small end and lies on neutral line, and larger stress acts on small end and nearby. Results of testing show that stress character in tension stress and compressive stresses are consistent with the calculation results from formula. The limit bending moment calculation results from formula of reducing elbow is conservative, but is consistent with the formula of the limit bending moment, which is controlled by small end of reducing elbow.

Exact Solutions for Elastoplastic Stress Distribution in Functionally Graded Curved Beams Subjected to Pure Bending

The material of the beam is presumed to be isotropic and ideally elastic-plastic and the elastic modulus and yield limit vary radially according to power law functions. Based on Tresca's yield criterion and its associated flow rule, the expressions of the stresses and displacements in four different stress states are presented. The condition to achieve the concurrent yielding on the inner and outer surfaces is given explicitly. The position of the neutral plane and the plastic limit bending moment in a completely plastic state are determined. The distinct characteristics of elastoplastic behavior of functionally graded materials (FGMs) curved beam are shown in examples.