Unknown Displacement Research Articles

First Principles Variational Formulation and Analytical Solutions for Third Order Shear Deformable Beams with Simple Supports using Fourier Series Method

The Fourier series method for solving bending problems of third-order shear deformable beams (TSBD) is presented in this paper. The theory accounts for transverse shear deformation and is suitable for moderately thick beams. Transverse shear stress-free conditions are valid at the beam surfaces for TSDB. The field equations are two coupled ordinary differential equations in terms of two unknown displacement functions – transverse deflection w(x) and warping function For simply supported ends considered, the loading and unknown functions w(x) and are represented in Fourier series that satisfies the boundary conditions. The problem is reduced to a system of algebraic equations in terms of the Fourier coefficients wn and which is solved to obtain wn and Axial and transverse displacements fields, axial bending stress and transverse shear stress are then determined for the cases of point load at midspan, uniformly and linearly distributed loads over the span. The results are identical with previous results obtained by other scholars who used Ritz variational methods and other mathematical tools. For moderately thick beams under uniform load with l/h = 4, the results obtained for is 0.516% greater than the exact result by Timoshenko and Goodier while for thick beams under uniform load with l/h = 2, the result for is 1.96% greater than the exact result by Timoshenko and Goodier. Similar acceptable variation was obtained for for beams under linearly distributed load with the variations being less than 2% for l/h = 2. However, unacceptable variations of 68.37% were found for in thick beams under point load, for l/h = 2.0. The variation for however reduced to 12.513% for moderately thick beams under point load for l/h =4.

Iterative Method of Determining Stress Intensity Coefficients Under Dynamic Loading of the Crack System

An elastic isotropic body in a state of plane deformation, which contains a system of randomly placed cracks under the action of a dynamic (harmonic) loading, is considered. The authors set the problem of determining the stress field around the cracks under the conditions of their wave interaction. The solution method is based on the introduction of displacements in the body in the form of a superposition of discontinuous solutions of the equations of motion constructed for each crack. With this in mind, the initial problem is reduced to a system of singular integro-differential equations with respect to unknown displacement jumps on the crack surfaces. To solve this system, a new iterative method, which involves solving a set of independent integro-differential equations that differ only in their right-hand parts at each iteration, is proposed. For the zero approximation, solutions that correspond to individual cracks under the action of dynamic loading are chosen. Such a new approach allows to avoid the difficulties associated with the need to solve systems of integro-differential equations of large dimensions that arise when traditional methods are used. Based on the results of the iterations, formulas for calculating the stress intensity coefficients for each crack were obtained. In the partial case of four cracks, a good agreement between the results obtained during the direct solution of the system of eight integro-differential equations by the mechanical quadrature method and the results obtained by the iterative method was established. In general, numerical examples demonstrate the convergence and stability of the proposed method in the case of systems with a fairly large number of densely located cracks. The influence of the interaction between cracks on the stress intensity factor (SIF) value under dynamic loading conditions was studied. An important and new result for fracture mechanics is the detection of the absolute maximum of the normal stresses at certain frequencies of the oscillating normal loading. The number of interacting cracks and the configuration of the crack system itself affect the values of the frequencies at which SIF reaches a maximum and the maximum values. These maximum values significantly (by several times) exceed the SIF values of single cracks under a similar loading. At the same time, under conditions of static or low-frequency loading, it is possible to reduce the SIF values compared to the SIF for individual cracks. When cracks are sheared, the values of the tangential stresses have a tendency to decrease with increasing frequency, and their values do not significantly differ from the values of the tangential stress for an individual crack.

Nonlinear temperature dependent and visco-elastic foundation effects on the free vibration of functionally graded sandwich plates with ceramic foam core

This paper investigates the vibrations of a functionally graded sandwich plate (FG sandwich plate) with a ceramic foam core resting on a viscoelastic foundation. The focus is on the influence of temperature changes on sandwich plates that have foam core and FG face sheet on the free vibration. The damping coefficient and different schemes of sandwich plate have been considered in the current work. The analysis uses a quasi-3D theory, which reduces the number of unknown displacements and simplifies the equations by incorporating integral terms. Two models are employed to represent the non-uniformity of the ceramic foam core. To account for the influence of the supporting layer, a viscoelastic foundation is included. The governing equations are derived using Hamilton’s principle and solved using the Navier solution method. The paper comprehensively discusses how the volume fraction index k, geometric properties, ceramic foam distribution, viscoelastic foundation, and different temperature rise profiles impact the vibration response of the FG sandwich plate.

Deriving Ritz formulation for the Static Flexural Solutions of Sinusoidal Shear Deformable Beams

This paper presents the Ritz variational method for the static bending analysis of sinusoidal shear deformable beams. The theory accounts for transverse shear deformation and satisfies transverse shear stress-free conditions at the top and bottom surfaces of the beam. The total potential energy functional for the thick beam bending problem is formulated and minimized using a Ritz procedure. The problem considered simply supported boundary conditions and two cases of loading – uniformly distributed load over the span and point load at the center. The function is a function of two unknown displacement functions constructed in terms of unknown generalized displacement parameters and shape functions that satisfy the boundary conditions. Ritz minimization of the functional is used to find the generalized displacement parameters and then the displacements w(x) and The displacements and stresses are found for the loading distributions considered. It is found that the results obtained agree remarkably well with the exact results obtained using the theory of elasticity. The differences between the present results and the exact solutions are less than 0.3% for maximum transverse displacement for both cases of loading considered. For uniform load over the beam, the result for l/t = 10 is 0.112% greater than the exact solution. For the point load at the center, the result for l/t = 10 is 4.468% greater than the exact solution. The increased difference in the point load case is due to the singular nature of the point load problem.

Drive-By Identification of Joint Bridge Damage and Surface Roughness Based on Sensitivity Analysis of Residual Contact Point Deflections

Drive-by inspection of bridge damage using a passing vehicle’s dynamic response has great potential in bridge damage identification. Among the current drive-by methodologies, the methods based on bridge modeling have been proposed for the quantitative identification of bridge damage and bridge surface roughness. However, the coupling of bridge damage and unknown surface roughness increases the difficulty of the identification problem and most previous studies conduct the identification task of bridge damage or bridge surface roughness separately. In this paper, a novel method is proposed for the identification of joint bridge damage and bridge surface roughness based on the residual bridge deflections from the front and rear wheels contact points of an instrumented passing vehicle. The vehicle–bridge interaction is analyzed using a half-car model of a four degrees of freedom (4-DOF) with front and rear vehicle wheels. First, the vehicle–bridge unknown forces and displacement of the vehicle axles are identified by the generalized Kalman filter under unknown input (GKF-UI) proposed by the authors. The sensors can be installed conveniently on the vehicle body due to GKF-UI. Then, the residual bridge deflections from the front and rear vehicle wheels contact points are utilized to eliminate the effect of road surface roughness. Afterward, bridge structural damage is identified based on the sensitivity analysis of residual bridge deflections from the front and rear contact points with [Formula: see text]1-norm regularization. Finally, bridge road surface roughness is estimated based on the identified unknown forces and updated bridge model with damage. The results of numerical identification examples demonstrate the effectiveness of the proposed method for the identification of joint bridge damage and surface roughness.

A reduced-dimension method for unknown Crank-Nicolson finite element solution coefficient vectors of elastic wave equation with singular source term

Herein, we mainly develop a new reduced-dimension method for the unknown finite element (FE) solution coefficient vectors to the elastic wave equation containing the known singular source term, the unknown displacement vector, and the unknown stress tensor matrix. For this end, we need first to develop the Crank-Nicolson (CN) FE (CNFE) method with unconditional stability, unconditional convergence, and second-order time accuracy for the elastic wave equation and analyze the existence, unconditional stability, and errors of the CNFE solutions. After that we employ a proper orthogonal decomposition to lower the dimensionality of the unknown FE solution coefficient vectors in the CNFE method for the elastic wave equation and to design a new reduced-dimension extrapolated CNFE (RDECNFE) method. Subsequently, we employ matrix analysis to discuss the existence, unconditional stability, unconditional convergence, and errors of the RDECNFE solutions. Finally, we use two numerical examples to show the superiority of the RDECNFE method and our theoretical results. It is also shown that the RDECNFE method is feasible and effective for solving the elastic wave equation.

Free and Forced Vibration Analysis of Carbon/Glass Hybrid Composite Laminated Plates Under Arbitrary Boundary Conditions

This paper presents the theoretical investigations on the free and forced vibration behaviours of carbon/glass hybrid composite laminated plates with arbitrary boundary conditions. The unknown allowable displacement functions of the physical middle surface are expressed in terms of standard cosine Fourier series and sinusoidal auxiliary functions to ensure the continuity of the displacement functions and their derivatives at the structural boundaries. Arbitrary boundary conditions are achieved through the introduction of an artificial spring technique. The first shear deformation theory and Lagrange equations are utilized to derive the energy expression, and the eigenvalue equations associated with free and forced vibration are obtained by Rayleigh-Ritz variational operations. Subsequently, these equations are then solved to determine the natural frequency, mode of vibration, and the steady-state displacement response under forced excitation. The new results are compared with those from references and finite element methods to verify the convergence, accuracy and efficiency of the analytical method. The effects of hybrid ratios, stacking sequences, lamination schemes, fibre orientation, boundary conditions and excitation force on the free and forced vibration behaviours of the carbon/glass hybrid composite laminated plates are analyzed in detail.

Analysis of Intact/Delaminated Composite and Sandwich Beams Using a Higher-Order Modeling Technique

A simple higher-order model (HOM) is presented in this study for the bending analysis of an intact or delaminated composite and sandwich beam. This model adopts the concept of sub-laminates to simulate multilayered structures, and each sub-laminate takes cubic variation for axial displacement and linear variation for transverse displacement through the thickness. A sub-laminate possesses displacement components at its surfaces (bottom and top) that provide a straightforward way to improve the accuracy of prediction by stacking several sub-laminates. Thus, analysts will have the flexibility to balance the computational cost and the accuracy by selecting an appropriate sub-lamination scheme. The proposed model was implemented by developing a C0 beam element that has only displacement unknowns. The model was used to solve numerical examples of composite and sandwich beams to demonstrate its performance.

A model for laminated composite beams based on a novel zig-zag theory

We develop a new model called C-311 for laminated composite beams based on a new zig-zag theory that is firstly proposed in this paper. The in-plane displacement field of the model includes two parts: part one is regarded as the global displacement field which is made of third-order polynomial about z, part two is the local displacement field which is the new zig-zag theory. By satisfying inter-laminar transverse stress continuity and the free surface conditions, the local displacement unknown variables can be eliminated, and the final displacement field should be expressed by the fixed number of unknown displacement variables. For testing the ability of C-311, the classical example about a simply supported laminated beam is investigated under mechanical loading in this paper. In comparison with other models of laminated composite beams, the numerical results show that the present model can give more accurate analysis, and are closer to the exact solutions. Especially the accurate transverse shear stresses are able to be directly obtained from the constitutive equations without any processing, and that is never done for other zig-zag models existing in the literature and benefit for practical engineering applications in the finite element method.

Robust SR-STAP algorithms in partly calibrated arrays for airborne radar

Sparse recovery based space-time processing (SR-STAP) techniques are capable of achieving satisfactory clutter suppression and target detection performance, even with limited training samples. The majority of existing approaches are developed under the assumption of ideal uniform linear arrays, where no imperfections are present. In this study, we consider the robust STAP problem in the context of airborne partly calibrated array (PCA) composed of several well calibrated subarrays with unknown inter-subarray gain-phase and displacement errors. Initially, the space-time signal for airborne STAP radar based on PCAs is modeled. Subsequently, the authors propose two robust SR-STAP algorithms, considering both grid-based and gridless methods, leveraging the block-sparsity and low rank structural characteristics inherent in the signal model. Extensive numerical experiments have shown the significant performance benefits including small sample applicability and resilience to inter-subarray gain-phase and displacement errors.

A third-order shear deformation plate bending formulation for thick plates: first principles derivation and applications

A third-order shear deformation plate bending formulation is presented in this study from the first principles. The derivation assumed a displacement field constructed using third-order polynomial function of the transverse (z) coordinate; and made to apriori satisfy the linear three-dimensional (3D) kinematics relations as well as the transverse shear stress free boundary conditions at the top and bottom plate surfaces. The formulation thus has no need for shear stress correction factors of the first-order shear deformation plate theories. The domain equations of equilibrium are obtained as a set of three coupled differential equations in terms of three unknown displacements. The system of coupled equations is solved for simply supported rectangular and square plates subjected to four cases of loading distributions: sinusoidal loading, uniformly distributed loading, linearly distributed loading and point load at the plate center. Navier’s double trigonometric series method is used to construct trial solutions for the three displacement functions such that the boundary conditions are satisfied identically. The integration problem is thus reduced to an algebraic problem and is solved for each considered loading. It is found that the present formulation gives exact results for the normal stresses σxx for sinusoidal and uniformly distributed loads. The study further showed that the results for deflection and stresses agreed with Krishna Murty’s higher order shear deformation plate theory results. The present formulation gave accurate results because of the inclusion of transverse normal strain effects in the formulation. The formulation gives a quadratic variation of the transverse shear stresses across the thickness in consonance with the theory of elasticity method.

DIRECT SOLUTION OF THE DYNAMICAL ELASTICITY PROBLEM FOR A QUARTER SPACE

The wave field of an elastic quarter space is constructed when one face is rigidly fixed and a dynamic normal compressive load is concentrated at the point on another face. The problem was solved by the direct application of the integral Laplace and Fourier transforms to the motion equations and the boundary conditions. This operation leads to the one-dimensional vector inhomogeneous boundary value problem with respect to unknown displacement’s transformants. The problem was solved using the matrix differential calculus. A fundamental matrix and a decreasing solution to the corresponding homogenous matrix equation were constructed with a basic residue theorem. A singular integral equation was obtained in the process by satisfying unrealized boundary condition. Weakly convergent part of the equation was summed up. The behavior of the unknown function had been analyzed based on its mechanical sense. The form of unknown function was expressed as a series on Laguerre polynomials. The original displacements’ field was found after an application the inverse integral transforms.

Porosity effects on the dynamic response of arbitrary restrained FG nanobeam based on the MCST

Abstract In this study, two different general eigenvalue problems for nanobeams made of functionally graded material with pores in their sections according to Rayleigh beam theory using modified couple stress theory are established. Fourier sine series and Stokes transformation are used for the solution. First, the partial differential equation of motion of the problem is discretized into an ordinary differential equation. Then, the Fourier sine series of infinite series is substituted into this ordinary differential equation to determine the Fourier coefficient. Using the force boundary conditions of the system, Stokes’ transformation is performed at both ends to include elastic spring parameters. The unknown displacement terms are discretized to form two eigenvalue problems. By solving these eigenvalue problems, vibration frequencies for different boundary conditions can be found analytically. The variations of some parameters are discussed in a series of graphs.

An explicit pseudo‐energy conservative scheme for contact between deformable solids

AbstractWe extend the work of Marazzato et al. [Int. J. Numer. Methods Eng, 121:5295‐5319] on elastodynamics to the treatment of contact. To that end, we propose adequate handling of boundary conditions, either through the resolution of local problems on each of the face displacement unknowns or through interpolation from nearest neighbours. Adapting the time‐integration strategy adopted in Marazzato et al. [Comput Methods Appl Mech Eng, 347:906‐927], it is possible to conserve both momentum and a pseudo‐energy exactly. Numerical results are presented to illustrate the accuracy of contact treatment and the energy conservation of the system.

Numerical modeling of nonlinear deformation processes for shells of medium thickness

When modeling a nonlinear isotropic eight-node finite element, the main kinematic and physical relationships are determined. In particular, isoparametric approximations of the geometry and an unknown displacement increment vector, covariant and contravariant components of basis vectors, metric tensors, strain tensors (Cauchy - Green and Almansi) and true Cauchy stresses in the initial and current configuration are introduced. Next, a variational equation is introduced in the stress rates in the actual configuration without taking into account body forces and the Seth material is considered, where the Almansi strain tensor is used as the finite strain tensor. Linearization of this variational equation, discretization of the obtained relations (stiffness matrix, matrix of geometric stiffness) is carried out. The resulting expressions are written as a system of linear algebraic equations. Several test cases are considered. The problem of bending a strip into a ring is presented. This problem is solved analytically, based on kinematic and physical relationships. Examples of nonlinear deformation of cylindrical and spherical shells are also shown. The method proposed in this paper for constructing a three-dimensional finite element of the nonlinear theory of elasticity, using the Seth material, makes it possible to obtain a special finite element, with which it is quite realistic to calculate the stress state of shells of medium thickness using a single-layer approximation in thickness. The obtained results of test cases demonstrate the operability of the proposed technique.

Bending Vibrations of an Elastic Rod Controlled by Piezoelectric Forces

Bending vibrations of a thin elastic rod of rectangular cross-section are studied. A number of piezoelectric actuators (elements) is symmetrically attached without gaps to two opposite sides of the rod. Each element is glued to the neighboring ones, forming with the rod a single elastic body in the form of a rectangular parallelepiped. The body is hinged at both ends relative to the cross-sectional axis parallel to the piezoelectric layers. In opposite piezoelements, homogeneous fields of normal stresses are set antisymmetrically as functions of time. These stresses are parallel to the axis of the rod and force the elastic system to perform bending motions. Within the framework of the linear theory of elasticity for the considered system, generalized formulations of the initial-boundary value problem and the corresponding eigenvalue problem are given. These problems are defined through unknown displacements and the time integrals of mechanical stresses. An approximation of the displacement and stress fields, which is polynomial in transverse coordinates, is proposed. This approximation exactly satisfies the homogeneous boundary conditions for stresses on the lateral sides and takes into account the symmetry properties of the bending motions. For the chosen approximation, the boundary value problem for eigenvalues is exactly solved. Two branches of eigenvalues are found and used to reduce the initial-boundary value problem to a countable system of first-order ordinary differential equations with respect to complex variables. The dynamical system is decomposed into independent infinite-dimensional subsystems with a scalar control input. One of these subsystems is not controllable. For the remaining subsystems, each corresponding to a pair of piezoelectric elements, a control law for vibration damping is proposed for a specific number of the lower modes associated with the lower branch.

Formulation for Multiple Cracks Problem in Thermoelectric-Bonded Materials Using Hypersingular Integral Equations

New formulations are produced for problems associated with multiple cracks in the upper part of thermoelectric-bonded materials subjected to remote stress using hypersingular integral equations (HSIEs). The modified complex stress potential function method with the continuity conditions of the resultant electric force and displacement electric function, and temperature and resultant heat flux being continuous across the bonded materials’ interface, is used to develop these HSIEs. The unknown crack opening displacement function, electric current density, and energy flux load are mapped into the square root singularity function using the curved length coordinate method. The new HSIEs for multiple cracks in the upper part of thermoelectric-bonded materials can be obtained by applying the superposition principle. The appropriate quadrature formulas are then used to find stress intensity factors, with the traction along the crack as the right-hand term with the help of the curved length coordinate method. The general solutions of HSIEs for crack problems in thermoelectric-bonded materials are demonstrated with two substitutions and it is strictly confirmed with rigorous proof that: (i) the general solutions of HSIEs reduce to infinite materials if G1=G2, K1=K2, and E1=E2, and the values of the electric parts are α1=α2=0 and λ1=λ2=0; (ii) the general solutions of HSIEs reduce to half-plane materials if G2=0, and the values of α1=α2=0, λ1=λ2=0 and κ2=0. These substitutions also partially validate the general solution derived from this study.

A Three-Stage Prediction Method for Track Displacement During Shield Tunneling

Track displacement is an important factor of track irregularity. Existing researches related with track displacement prediction generally ignore the influence from underground construction engineering such as shield tunneling, resulting in inaccurate estimation of track displacement. To fill this gap, we propose a three-stage framework to predict the track displacement when the shield tunnel under crosses the existing tunnel. Firstly, by considering the curved shield tunneling, a three-dimensional model is constructed to estimate the total ground displacement during the whole tunneling process. Furthermore, the soil-tunnel interaction model is established to estimate the deformation of the existing tunnel. To tackle the issue of unknown node displacements, cubic splines are used to interpolate the unknown values of tunnel displacements. On this basis, the direct stiffness method is used to estimate the track displacement and to calculate the track irregularity. Finally, the effectiveness of the proposed method is verified and the prediction performance on the track irregularity is evaluated using a real engineering case and the finite element simulation. The main contributions of this article lie in the modeling of the curved scenario for the estimation of the ground loss, as well as the combination of cubic splines and direct stiffness method, which improve the accuracy of the track displacement estimation during shield tunneling.

A multiple-data-based direct method for inverse problem in three-dimensional linear elasticity

Estimating the unknown Young’s modulus and boundary conditions of an elastic object from locally observed boundary data is an inverse problem which is usually solved by iterative methods. In this paper, a multiple-data-based direct (MD) method is proposed to solve this inverse problem without iterations. The proposed method applies a global normalized objective function and a global regularization coefficient τg for energy-like regularization to ensure the convergence of the unknown displacement boundary conditions. A coarse-mesh based preprocessing algorithm is proposed to speed up the determination of the regularization coefficient. A multi-force-position method is proposed to obtain observation data, which improves accuracy of the MD method when the number of experiments is limited. Both numerical and physical experiments were conducted. Compared with the single-data-based direct (SD) methods proposed in our previous work, the MD method yields more accurate estimation of Young’s modulus and boundary conditions in both numerical and physical experiments.

Isogeometric Analysis for Free Vibration of Functionally Graded Plates Using a New Quasi-3D Spectral Displacement Formulation

This paper presents a novel quasi-three-dimensional shear deformation theory called the spectral displacement formulation (SDF) for analyzing the free vibration of functionally graded plates. The SDF expresses the unknown displacement field as a unique form of the Chebyshev series in the thickness direction. By increasing the truncation number in the Chebyshev series, the bending analysis results can approach the three-dimensional elasticity solution and satisfy the traction-free boundary conditions without requiring a shear correction factor. The SDF is an extension of the classical plate theory, thereby naturally avoiding the shear-locking phenomenon. These characteristics enable the SDF to apply to plates of arbitrary thickness while maintaining accuracy. The nonuniform rational B-spline-based isogeometric approach is employed to enhance the applicability of this theory to free vibration analysis of functionally graded plates with complex geometries and different boundary conditions. Numerical examples are presented to demonstrate the accuracy and reliability of the proposed method in analyzing the free vibration of functionally graded plates.