A variational method for calculating the longitudinal deformation of a shield tunnel in soft soil caused by grouting under tunnel

Purpose The paper aims to predict longitudinal deformation of a tunnel caused by grouting under the tunnel bottom in advance according to the grouting parameters, which can ensure the safety of the tunnel structure during the grouting process and also help to design the grouting parameters. Design/methodology/approach The paper adopted the analytical approach for calculating the longitudinal deformation of a shield tunnel caused by grouting under a tunnel, including usage of the Mindlin’s solution, the minimum potential energy principle and case validation. Findings The paper provides a variational method for calculating the longitudinal deformation of a shield tunnel in soft soil caused by grouting under the tunnel, which has high computational efficiency and accuracy. Originality/value This paper fulfils an identified need to study how the longitudinal deformation of a shield tunnel in soft soil caused by grouting under the tunnel can be calculated.

Piezoelectrically induced snap-through buckling in a buckled beam bonded with a segmented actuator

In this article, a mathematical model is developed to analyze piezoelectrically induced deformations of bistable smart beams with various geometric configurations and boundary conditions. The model delineates a straight beam bonded with a segmented piezoelectric material. Three types of support conditions are considered, namely, simple, clamped–clamped, and clamped–pinned supports. The beam is buckled into two possible stable curved shapes by means of edge shortening compression. A sudden change in transverse deflection from one to the other stable shape, so-called snap-through action, can be stimulated by electrical activation given to the piezoelectric material. The minimum potential energy principle associated with the modified Ritz method is used to predict developing shapes of the smart beam and the snap-through voltage. Principal minors of bordered Hessian matrix are calculated to determine the stability of the shapes obtained. Experiments were also conducted to corroborate the computational results in the case of the simply-supported smart beam. Comparisons between the two approaches reveal very good agreements in both mid-span displacements of buckled shapes and snap-through voltages. Finally, size and location of the segmented piezoelectric actuator are varied to search for the minimum critical electrical field for each support condition.

Interpolating Smoothed Particle Method for Elastic Axisymmetrical Problem

The interpolating reproducing kernel particle method is a meshless method with discrete points interpolation character. Coupling this method with the minimum potential energy principle of space axisymmetrical problems of elastic mechanics, the interpolating smoothed particle method (ISPM) is formed. The ISPM, which is a meshless method with discrete points interpolation character, can refrain from quadric error of fitting calculation in stress post-processing by obtaining global domain continuous stress fields directly. This method not only has the advantage in directly exerting boundary conditions just like the finite element method, but is also a new numerical method which has greater computational efficiency and precision than it in solving space axisymmetrical problems of elastic mechanics. Numerical examples are given to show the validity of the new meshless method in the paper.

Natural Frequency and Mode Shapes of Exponential Tapered AFG Beams on Elastic Foundation

A displacement based semi-analytical method is utilized to study non-linear free vibration and mode shapes of an exponential tapered axially functionally graded (AFG) beam resting on an elastic foundation. In the present study geometric nonlinearity induced through large displacement is taken care of by non-linear strain-displacement relations. The beam is considered to be slender to neglect the rotary inertia and shear deformation effects. In the present paper at first the static problem is solved through an iterative scheme using a relaxation parameter and later on the subsequent dynamic analysis is carried out as a standard eigen value problem. Energy principles are used for the formulation of both the problems. The static problem is solved by using minimum potential energy principle whereas in case of dynamic problem Hamilton’s principle is employed. The free vibrational frequencies are tabulated for exponential taper profile subject to various boundary conditions and foundation stiffness. The dynamic behaviour of the system is presented in the form of backbone curves in dimensionless frequency-amplitude plane and in some particular case the mode shape results are furnished.

Geometric nonlinear free vibration of axially functionally graded non-uniform beams supported on elastic foundation

In the present study non-linear free vibration analysis is performed on a tapered Axially Functionally Graded (AFG) beam resting on an elastic foundation with different boundary conditions. Firstly the static problem is carried out through an iterative scheme using a relaxation parameter and later on the subsequent dynamic problem is solved as a standard eigen value problem. Minimum potential energy principle is used for the formulation of the static problem whereas for the dynamic problem Hamilton’s principle is utilized. The free vibrational frequencies are tabulated for different taper profile, taper parameter and foundation stiffness. The dynamic behaviour of the system is presented in the form of backbone curves in dimensionless frequency-amplitude plane.

Boundary Conditions: A Key Factor in Tubular-String Buckling

Summary Boundary conditions for tubular-string buckling are divided into two kinds on the basis of the virtual work of the nonaxial force on the boundary conditions of the tubular string—namely, the first and the second kinds of boundary conditions. The first kind of boundary conditions means that the virtual work of no-axial force is zero, and both the conventional pinned and fixed ends belong to this kind. The second kind of boundary conditions means that the virtual work of no-axial force is not zero. Previous studies on tubular-string buckling mainly focus on the first kind but ignore the second kind of boundary conditions. In this paper, the effects of the two kinds of boundary conditions on tubular-string buckling are analyzed. The deflection of a long tubular string constrained in a wellbore is divided into the full helical-buckling section and transition section, whereas the transition section is divided further into the no-contact section and perturbed-helix section. The qualitative corresponding relation between boundary conditions and tubular-string buckling in transition and full helical-buckling sections is clarified. To clarify the quantitative relation between boundary conditions and tubular-string buckling, a general-packer model is proposed to depict the two categories of boundary conditions. With the general-packer model, the general potential energy of the tubular string is deduced. According to the minimum-potential-energy principle, the existence and stability of full helical-buckling solutions are given. The deflections of the tubular string in the no-contact and perturbed-helix sections are deduced with buckling differential equation and beam-column model. The bending moment, shear force, and contact force on the tubular string caused by buckling are also analyzed. The results show that boundary conditions, especially the second kind of boundary conditions, are an important factor that makes the tubular-string buckling problem complex, and this paper provides one source for a deeper understanding of boundary conditions.

Theoretical and Experimental Investigations on Shear Lag Effect of Double-Box Composite Beam with Wide Flange under Symmetrical Loading

ABSTRACTA new type of steel-concrete composite beam with double-box cross-section is proposed in this paper. In order to investigate stress behaviors and deflection characteristics of such composite beam with wide flange considering the shear lag effect, theoretical analysis and experimental study are launched simultaneously. Based on the minimum potential energy principle, governing differential equations in view of the shear lag effect are deduced by energy variational method, and analytical solutions of it's stress and deflection under the effect of symmetrical loading are calculated. The preceding analyses show that relative error is less than 14.71%, with a good agreement, and farther show that this method of theoretical derivation, which is used for analyzing shear lag effect of composite beam with wide flange, has certain reference and guidance.

Implement of Ameliorated ACM Element in Numerical Manifold Space for Tackling Kirchhoff Plate Bending Problems

Ab ridging the chasm between the prevalent ly employed continuum methods (e.g. FEM) and discontinuum methods (e.g. DDA) ,the numerical manifold (NNM) ,which utilizes two covers, namely the mathematical cover and physical cover , has evinced various advantages in solving solid mechanic al issues. The forth-order partial elliptic differential equation governing Kirchhoff plate bending makes it arduous to establish the -regular Lagrangian partition of unity ,nevertheless, this study renders a modified conforming ACM manifold element , irrespective of accreting its cover degrees, to resolve the fourth-order problems. In tandem with the forming of the finite element cover system that erected on r ectangular mesh es , a succession of n umerical manifold formulas are derived on grounds of the minimum potential energy principle and the displacement boundary conditions are executed by penalty function methods. The numerical example elucidates that , compared with the orthodox ACM element , the proposed methods bespeak the accuracy and precipitating convergence of the NMM .