Deflection Differential Equation Research Articles

Nonlocal strain gradient-based geometrically nonlinear vibration analysis of double curved shallow nanoshell containing functionally graded layers

It is a fact that while examining the mechanical behavior of nano/microscopic materials, size effects become apparent. This study aims at presenting a new approach to geometrically nonlinear vibration analysis of double curved shallow nanoshell on Kerr foundation containing functionally graded layers under the impacts of harmonic loading in the framework of nonlocal strain gradient theory and the classical shell theory. The three-parameter Kerr model is founded on the independence of the upper and lower elastic layers related to the shear layer. The motion equations of the double curved shallow nanoshell are deduced by introducing the Von Kármán strain-displacement relations. Governing equations are solved by displacement function method and Galerkin method to achieve the deflection differential equation, the fourth-order Runge-Kutta methods are implemented to solve the deflection differential equation of the dynamic system. In this study, the obtained outcomes are compared with other publications indicating that the theoretical results agree with the previously published ones. Afterward, the article investigates the effects of the material length scale parameter, the nonlocal parameter, the geometric parameters and the material properties on the geometrically nonlinear vibration analysis of the double curved shallow nanoshell containing functionally graded layers. This research paper can provide extensive references and valuable guidelines for the impact and application of double curved shallow nanoshell structures containing functionally graded layers.

Accurate identification of cross-sectional bending stiffness in large-scale wind turbine blades through static loading test

To ensure the safe and stable operation of wind turbines, it is essential that the blades have sufficient bending stiffness to prevent excessive deflection and tower impact. A method combining static loading test and numerical analysis is proposed to solve the problem of identifying the cross-sectional bending stiffness of actual blades. Two working conditions, namely multi-point static test and single-point static calibration, are considered. A mathematical model of a cantilever beam with an initial small curvature and variable cross section is established. By fitting the deflection measurement data obtained from the static loading test, the deflection differential equation is solved by the finite difference method to obtain expressions for the cross-sectional bending stiffness under large and small deflection. The analysis results of several blade models show that the bending stiffness identification error for most blade sections is less than 10%, confirming the effectiveness of this method. This provides a practical and theoretical basis for the design, analysis, and refinement of test parameters for large-scale blades.

Time dependent analysis of lateral bearing capacity of reinforced concrete piles combined with corrosion and scour

This study investigates the effect of chlorine corrosion and flow scour on the lateral bearing capacity of piles. A chloride diffusion model was established to obtain the initial corrosion time of reinforcement based on Fick's second law. The degradation of stiffness after corrosion is represented by a reduction coefficient. The finite difference method is used to solve the deflection differential equation of laterally loaded piles. The maximum scouring depth of piles in cohesive soil and the variation of scouring depth with time are calculated by empirical model. The calculated results from the analytical solution revealed that the pile stiffness decreases with the exposure time. As the time increase, the lateral displacement and shear force decrease on the other hand the bending moment increases. The increase of flow velocity and pile diameter results in the increase of the maximum scour depth around the pile. The bending moment, shear force, and lateral displacement of piles after scour increase significantly compared to the lateral bearing performance of piles before scour.

Mechanical Response Analysis for an Active–Passive Pile Adjacent to Surcharge Load

Due to the complexity of pile–soil interaction, there is little research on active–passive piles that bear the pile-top load transmitted from the superstructure and the pile shaft load caused by the lateral soil movement around the pile simultaneously. The purpose of this study is to analyze the displacement and internal force of active–passive piles. Most of the pile design codes in China use the elastic resistance method to describe the relationship between the lateral soil resistance and the horizontal displacement of the pile, but this is not accurate enough to analyze the internal force and deformation of the pile when the pile displacement is large. For this case, the passive load on the pile shaft caused by the adjacent surcharge load can be described in stages, and the p–y curve method can be used to express the relationship between the lateral soil resistance and the horizontal displacement of the pile. Additionally, taking both the active load (vertical force, horizontal force, and bending moment on the pile top) and the passive load into account, the deflection differential equation of the pile shaft is herein established, and a corresponding finite difference method program is implemented to obtain the calculations pursuant to the equation. The correctness of the analysis method and program was verified by two test cases. The results show that our calculation method can effectively judge the flow state of the soil around piles and accurately reflect the nonlinear characteristics of pile-soil interaction. Moreover, the influence depth of the pile displacement under the passive pile condition caused by the adjacent load is significantly greater than that under active pile condition, and the maximum pile-bending moment appears near the interface of soft and hard soil layer.

Analysis for laterally loaded pile behavior on offshore sand slope crest considering friction effect and variational base angle in FW

The nonlinear response of horizontally loaded piles located at the crest of offshore cohesionless soil slope is analyzed, based on the failure wedge model (FW) with variational base angle, in this study. The pile deflection differential equation considering the three-dimensional friction effect is derived; the failure wedge model with the variational base angle on slope crest is proposed; the three-dimensional friction effect along the pile on the failure wedge is considered; the calculation formula for ultimate soil resistance around pile on cohesionless soil slope crest is derived. The feasibility of the proposed method is proved by comparing the theoretical results with the test results. Finally, the slope effect and friction effect along the pile on laterally loaded pile behavior and the influence of pile diameter on the friction effect are analyzed and discussed based on the proposed method. Slope can weaken the soil resistance around the pile, while friction can strengthen that around the pile. The friction effect gets more significant with the increase of pile diameter.

A load-slip model for stud connector in steel-concrete composite structures

A new theoretical calculation method for the load-slip curve and the shear stiffness of the stud based on Winkler’s elastic foundation beam theory and deflection differential equation of the stud is proposed. Different stress characteristics of the elastic and plastic zones of the concrete are considered in the proposed method. The accuracy of the theoretical calculation method is validated by the test results. Moreover, the effect of concrete strength, stud diameter, and yield strength on the stud load-slip curve is analyzed. The results show that the proposed model can effectively reflect the change process of the load-slip curve of the test specimen stud connector. The stud diameter, stud yield strength, and concrete strength affect the peak slip, shear stiffness, and shear capacity of stud shear connectors. With an increase in stud diameter, the peak slip, shear stiffness, and shear bearing capacity of the stud are all increased. An increase in the yield strength of the stud increases its elastic slip, peak slip, and shear capacity. Lastly, with an increase in concrete strength, the elastic slip and peak slip of the stud both decrease, whereas the shear stiffness and shear bearing capacity are increased.

Analytical method for the out-of-plane buckling of the telescopic boom with guy cables.

The guy cables can effectively improve the bearing capacity for the telescopic boom of the crane, while the introduction of tensioned cables makes the buckling analysis complicated. The primary objective of this study was to propose an analytical method for the out-of-plane buckling of the telescopic boom with the spatial symmetric guy cables. To analyze the influence of the guy cables on the out-of-plane buckling property of the telescopic boom, the deflection differential equation of the multi-stepped telescopic boom with guy cables was established based on the theory of elastic beam, then the buckling characteristic equation to determine the critical load of the telescopic boom was derived. Comparison of results with that given by the finite element method showed the high accuracy of the proposed method. In the end, with this equation, the influences of the structural geometric parameters on the critical load were investigated. The results indicated that, in the engineering application, the critical load of the telescopic boom can be increased by decreasing the length ratio a/L or increasing the angle φ between the two cables. The influence of angle θ on the out-of-plane buckling analysis of the telescopic boom can be neglected. With the proposed method, the buckling behavior of the telescopic boom with guy cables can be solved accurately. The present work is significant to structural design and safety analysis of the telescopic boom, and it can be utilized to provide technical support for the structural design of telescopic boom.

Overall stability analysis of combined boom with telescopic main boom and fixed flying jib

On the purpose of accurately analyze the overall stability of the combined boom, and it also provides calculation basis for the structural design of the crane combined boom and the revision of the Chinese Design Rules for Cranes GB3811-2008. In order to analyse the overall stability of the combined boom accurately, the deflection differential equations of combined boom is established based on the vertical and horizontal bending theory. What’s more, combining the constraint and displacement boundary condition, the precise recursive expression of the buckling characteristic equations of the combined boom is derived. Taking the combined boom of 6-section telescopic main boom and fixed flying jib as an example, the proposed method and the ANSYS simulation method are used to calculate the critical bucking force of the combined boom, and the comparison results show that: the deduced instability characteristic equation of the combined boom is correct and accurate, which is necessary to the designing analysis in engineering.

Migration Law of the Roof of a Composited Backfilling Longwall Face in a Steeply Dipping Coal Seam

The artificial-caved rock composited backfilling approach can effectively restrain the dynamic phenomena in the coal seam and the associated roof and floor during mining operations, and can also improve the stability of the system of support and surrounding rock. In this study, based on the analysis of influencing factors affecting the surrounding rock movement and deformation of the composited backfilling longwall face in a steeply dipping coal seam, the roof mechanical model is developed, and the deflection differential equation is derived, to obtain the roof damage structure and the location of the roof fracture for the area without backfilling. The migration law of the roof under different inclination angles, mining depths, working face lengths, and backfilling ratios are analyzed. Finally, mine pressure is detected in the tested working face. Results show that the roof deflection, bending moment, and rotation drop with the increase of the inclination angle and backfilling ratio, whereas these parameters increase with greater mining depth and working face length. The roof failure location moves toward the upper area of the working face as the inclination angle and working face length increases, while it moves toward the center of the non-backfilling area with greater mining depth and backfilling ratio. Results from the proposed mechanical model agree well with the field test results, demonstrating the validity of the model, which can provide theoretical basis for a safe and efficient mining operation in steeply dipping coal seams.

Research on Buckling State of Prestressed Fiber-Strengthened Steel Pipes

The main restorative methods of damaged oil and gas pipelines include welding reinforcement, fixture reinforcement and fiber material reinforcement. Owing to the severe corrosion problems of pipes in practical use, the research on renovation and consolidation techniques of damaged pipes gains extensive attention by experts and scholars both at home and abroad. The analysis of mechanical behaviors of reinforced pressure pipelines and further studies focusing on “the critical buckling” and intensity of pressure pipeline failure are conducted in this paper, providing theoretical basis to restressed fiber-strengthened steel pipes. Deformation coordination equations and buckling control equations of steel pipes under the effect of prestress is deduced by using Rayleigh Ritz method, which is an approximation method based on potential energy stationary value theory and minimum potential energy principle. According to the deformation of prestressed steel pipes, the deflection differential equation of prestressed steel pipes is established, and the critical value of buckling under prestress is obtained.

Effects of surface drilling grouting on deformation properties of thin rock stratum above goaf

Strata movement of coal mining for surface drilling grouting of overlying loose layers is distinctly different from that of traditional caving mining. In order to calculate strata movement accurately after full-size grouting, based on the shear transfer mechanism along foundation horizontal, a mechanical model of strata movement was established using the theory of Pasternak double-parameter beam on elastic foundation, and the deflection differential equations of rock beam were derived. Then the influence mechanisms of varied grouting parameters were discussed and the results showed that, the maximum deformation change value of roof increase non-obviously with the increasing of elastic modulus ratio, while decrease non-linearly with the grouting thickness. Thus can provide theoretical reference for strata displacement control in the same or similar projects.

The influence of shear strain on critical load bearing capacity. Determination of instability factor $$ \varvec{k}_{\varvec{c}}^{\varvec{G}} $$ k c G

The paper is concerned with timber constructions. The paper presents a solution of a rod deflection differential equation. The rod deflection equation describes a compressed rod to which a transverse force is applied. The equation accounts for the influence of shearing forces on the magnitude of displacement. Based on the equation, the critical force and the rod deflection function at the rod’s initial curvature defined by a sinusoid were determined. Having found the rod deflection function and the critical force, a formula for the instability factor $$ k_{c}^{G} $$ was derived to account for the influence of shear strain. The paper demonstrates in which particular cases it is necessary to take into account the influence of deflection caused by shearing on the critical load bearing.

From the Euler–Bernoulli beam to the Timoshenko one through a sequence of Reddy-type shear deformable beam models of increasing order

Abstract A sequence of elastic Reddy-type shear deformable beams of increasing (odd) order is envisioned, which starts with the Euler–Bernoulli beam (first order) and terminates with the Timoshenko beam (infinite order). The kinematics of the generic beam, including the warping mode of the cross sections, is specified in terms of three deformation variables (two curvatures, one shear angle), work-conjugate of as many stress resultants (two bending moments, one shear force). The principle of virtual power is used to determine the (static) equilibrium equations and the boundary conditions. The equations relating the bending moments and shear force to the curvatures and shear angle are also reported. Two governing differential equations (equivalent to a deflection differential equation of the sixth order) are provided, along with their general solution to within six unknown constants; the solutions for the Euler–Bernoulli and Timoshenko beams are recovered for first and infinite orders, respectively. A principle of minimum total potential energy is shown to hold for the generic beam. Extended forms of the classical Navier formula for the computation of the (axial) normal stress, and of the Jourawski formula for the computation of the “equilibrium” shear stress, are provided. A simple cantilever beam is used for an illustrative application.

Out-of-Plane Stability Analysis of Crane Jib with Auxiliary Bracing

The analysis model is built to investigate the out-of-plane stability of crane jib with auxiliary bracing. Considering the second-order effect, the analytical expression of the out-of-plane buckling characteristic equation for the crane jib with auxiliary bracing is obtained by establishing the bending deflection differential equation of jib under the instability critical state with the method of differential equation. The equilibrium equation of the cable converging point in the lateral direction is introduced to solve the differential equation besides the boundary conditions and deformation compatibility equations. With the characteristic equation, the critical compression or the critical lifting load can be easily obtained. The characteristic equation is analytically expressed and the analytical results obtained agree very well with the finite element method (FEM) results. The validity of the characteristic equation is verified.

Out-of-Plane Stability Analysis for Crane Jib with Single Cable Considering Lateral Flexibility of the Cable Fixed Joint

The out-of-plane stability of crane jib is studied considering the lateral flexibility of the fixed joint. The analytical expression of the out-of-plane buckling characteristic equation for the crane jib with single cable is obtained by establishing the bending deflection differential equation of jib under the instability critical state with the method of differential equation. The equilibrium equation of the fixed point in the lateral direction is introduced to solve the differential equation besides the boundary conditions. The analytical results obtained agree very well with the finite element method (FEM) results. To consider the lateral flexibility of the cable fixed joint, a dimensionless stiffness coefficient measuring the lateral constraint was introduced to derive the out-of-plane buckling characteristic equation. The degeneration forms of the characteristic equation under the limit cases of zero lateral stiffness, infinite lateral stiffness are further discussed. And the influence of the lateral stiffness of fixed joint on the stability of jib is investigated. It is shown that the increase of the lateral stiffness will significantly improve the buckling load of the crane jib especially when the lateral stiffness is very small.

Out-of-the-Lifting-Plane Stability Analysis of Crane Telescopic Boom with Cable Exerted at the Top

To enhance the carrying capacity of the crane variable cross-section telescopic boom, the usual practice is using the cable at its top end, it makes the out-of-the-lifting plane stability problem of crane telescopic boom become solving the Euler critical force with follower force. This paper established the deflection differential equations of crane telescopic boom model which under actions of cable, with proper boundary conditions, the recurrence formula of buckling characteristic equations were presented, and some practical applications were given. The influence on buckling critical force of crane boom due to the ratio of the length of cable and crane boom was discussed. Took certain four-sectioned telescopic boom as example, the destabilizing critical force was calculated, the result showed that in comparison with the ANSYS method, the buckling characteristic equations in this paper is completely correct.

Precise Stability Analysis of Stepped Telescopic Booms and Practical Algorithm

To analyze the stability of stepped telescopic booms accurately, using vertical and horizontal bending theory, this paper established the deflection differential equations of stepped column model of arbitrary sectioned telescopic boom, the stability were analyzed, and obtained the precise expression of the buckling characteristic equation; Took certain seven-sectioned telescopic booms as example, by comparing the results with ANSYS, the accuracy of the equations deduced in this paper was verified. Presented the equivalent component method for the stability analysis of multi-stepped column, the equivalent cross-sectional moment of inertia was deduced by energy method, thus the stability of stepped column equivalent to that of constant section component. By comparing the results with exact value, the precision of equivalent component method was verified which was convenient for stability analysis of telescopic boom.

Analysis of Flexibility and Stability of Crane Telescopic Boom with Elastic Restraint and Second-Order Effect

The cylinder support crane telescopic booms deformation and stability analysis model in the lifting plane is equivalent with the multistep column with elastic restraint. To analyze the lateral flexibility and vertical stability of the telescopic booms with elastic restraint accurately, this paper established the deflection differential equations of multi-sectioned telescopic booms with second-order effect, introduced proper boundary conditions, obtained the precise recurrence lateral deflection differential equations and the buckling characteristic equations of arbitrary sectioned telescopic booms, and some practical applications of the buckling characteristic equations were presented. Took certain five-sectioned telescopic booms as example, by comparing the results with ANSYS method, the accuracy of the equations deduced in this paper was verified.

Accurate Calculation about Euler Critical Force of Multistep Telescopic Cylinder with Two Different Kinds Support Conditions

Variable cross-section telescopic cylinder is the frenquently-used member being loaded, which the buckling critical force is variable as the boundary and load condition changed. According to two common boundary supprot conditions, this paper established the deflection differential equations of arbitrary multistep telescopic boom by vertical and horizontal bending theory with second-order effect, introduced the boundary condition and deformation compatibility condition, obtained the recurrence expression of buckling characteristic equations, and some practical applications were presented. Took certain four-sectioned telescopic cylinder as example, calculated the buckling critical force by the method and compared the results with ANSYS, the accuracy of the buckling characteristic equations deduced in this paper were verified.

Stability Analysis for Horizontal Boom with Double Pulling Rods under Non-Conservative Forces

The horizontal boom of the tower crane in the rotary plane is a cantilever structure with large flexibility. Its stability analysis belongs to critical load problems under double non-conservative forces. In this paper, the second-order theory is used to establish the deflection differential equation in the rotary plane of the horizontal boom. According to the deformation compatibility conditions and the boundary conditions, the exact expression of the deflection equation and the instability characteristic equation in the rotary plane are obtained. The premise of the stability analysis is to determine the force relationship of the two pulling rods in the horizontal boom with double lifting points, which is a statically indeterminate structure. This paper is aimed at the structural characteristics and the load forms of the general boom. For the calculation of the pulling rod forces in the boom, the general expression is deduced considering the variety of the cross-section, the load and the position of lifting points. A highly efficient method for the stability analysis of the horizontal boom with double pulling rods is provided by the analytic expression of the instability characteristic equation and the internal forces in the statically indeterminate pulling rods.