In-plane Displacement Distributions Research Articles

Buckling analysis of the porous sandwich functionally graded plates resting on Pasternak foundations by Navier solution combined with a new refined quasi-3D hyperbolic shear deformation theory

This article presents a numerical method for analyzing the compressive buckling of porous power-law functionally graded (PPFG) plates placed on Pasternak foundations, which is based on a novel refined quasi-3-dimensional hyperbolic shear deformation theory (RQHSDT) combined with the Navier solution technique. The RQHSDT employs a novel hyperbolic distribution of in-plane displacements through the plate thickness to account for non-linear variations in transverse shear stresses, satisfies the traction free boundary conditions on the top and bottom surfaces, and accounts for shear deformation and thickness stretching effects of the plates without the use of any shear correction factors. Unlike classical shear and normal deformation theories, the RQHSDT has only four variables in its displacement field, as opposed to five or more variables in other quasi-3D shear deformation theories, which reduces the number of unknown variables and governing equations in plate analyses and simplifies calculations. The governing equations of the buckling problem are derived using the virtual work principle. In problem solving, the Navier solution technique is used to obtain the exact solutions for the rectangular PPFG plates with simply supported edges subjected to in-plane loading. The accuracy of the proposed approach is confirmed by comparing the obtained results to those of previously published quasi-3D shear deformation theories. Using comprehensive parametric studies, the effects of gradient index, porosity fraction index, porosity distribution type, geometric parameters, two-direction compression ratio, and stiffness of foundation parameters on buckling of PPFG plates are investigated.

Geometrically nonlinear dynamic analysis of laminated composite plate using a nonpolynomial shear deformation theory

A computationally efficient C0 finite element model in conjunction with the nonpolynomial shear deformation theory (NPSDT) is extended to examine the free and forced vibration behavior of laminated composite plates. The employed NPSDT assumes the nonlinear distribution of in-plane displacements which qualify the requirement of traction free boundary conditions at the top and bottom surfaces. The present formulation utilizes both von Kármán and Green–Lagrange type of strain–displacement relations to model the geometric nonlinearity. Using Hamilton’s principle, the nonlinear governing equation of motion is derived and then discretized based on the nine-noded Lagrange element. The obtained equations are solved by utilizing unconditionally stable Newmark’s scheme in conjunction with Newton–Raphson method. A damping effect in the transient analysis has been introduced in the framework of the Rayleigh damping model. The steady state forced vibration analysis has also been carried out by employing harmonic force with excitation frequency around the natural frequency. The arc-length continuation method is applied to obtained the frequency response. The present model has been validated for a wide range of problems and a detailed numerical study has been carried out for several types of boundary conditions under various types of loading with different magnitude of the load.

Layerwise Theories of Laminated Composite Structures and Their Applications: A Review

It is very challenging to accurately analyze the displacement and stress fields in the laminated composite structures due to the transverse anisotropy and higher transverse flexibilities. The equivalent single-layer theory (EST) is first developed to simplify this complex 3D problem to a pure 2D problem based on a displacement assumption in the thickness direction. The EST gives good results for the global responses of the very thin laminates with minimum computation cost, but poor results for the local responses at the interfaces and can not presents the zig-zag distribution of in-plane displacements. The elasticity solutions based on the 3D displacement-based finite element method (FEM) can present the accurate displacement and stress fields, but requires huge computational cost. As a quasi 3D method, the layerwise theories (LWT) is more accurate than most of the ESTs, and its computational cost is less than that of the 3D-based displacement FEMs, so it is attracting more and more attention with the rapid development of computer technology. This paper presents an overall review for the LWTs of the laminated composite structures and their applications. In general, there are two basic schemes employed to establish a LWT according to the construction of the displacements fields in the thickness direction. In the first scheme, the laminated composite structures are described as an assembly of individual layers, and these individual layers are combined by the interlaminar relationships to keep the displacement continuity and stress equilibrium. All of the individual layers are simulated by using the EST. In the second scheme, the displacement and/or stress fields along the thickness direction are constructed by a 1D interpolation functions and the in-plane displacement and/or stress fields are discretized by the 2D finite elements. The review in this paper is organized by the this classification.

Accurate full-field optical displacement measurement technique using a digital camera and repeated patterns.

In this study, a novel, fast, and accurate in-plane displacement distribution measurement method is proposed that uses a digital camera and arbitrary repeated patterns based on the moiré methodology. The key aspect of this method is the use of phase information of both the fundamental frequency and the high-order frequency components of the moiré fringe before and after deformations. Compared with conventional displacement methods and sensors, the main advantages of the method developed herein are its high resolution, accuracy, speed, low cost, and easy implementation. The effectiveness is confirmed by a simple in-plane displacement measurement experiment, and the experimental results indicate that an accuracy of 1/1000 of the pitch can be achieved for various repeated patterns. This method is useful for various applications ranging from the study of displacement and strain distributions in materials science, the biomimetics field, and mechanical material testing, to secure the integrity of infrastructures.

Dynamic Thermal Deformation Measurement of Large-Scale, High-Temperature Piping in Thermal Power Plants Utilizing the Sampling Moiré Method and Grating Magnets

Fast and accurate deformation measuring techniques show promising potential for monitoring high-temperature piping and other machinery in the energy sector, such as in thermal, steam, or nuclear power plants. Optics-based, nonintrusive, and real-time monitoring systems are critical for preventing high-temperature steam leaks, unexpected accidents, explosions, and other failures that can have catastrophic consequences for the power supply, the workers, and the public. In this study, a dynamic thermal deformation measurement technique utilizing the sampling Moire method and grating magnets is developed for large-scale, high-temperature piping in a thermal power plant. Because of the high temperature of the piping of nearly 300 °C several 100 mm by 100 mm ferrite magnets were placed to select locations of the high-temperature piping. In each ferrite magnet, a regular, two-dimensional binary grating pattern with a pitch of 15 mm was painted on the surface, and the grating magnets were used as the reference grating to analyze the small in-plane displacement distribution using the sampling Moire method. A compensation method of misalignment angle is also proposed to improve the measurement accuracy. The effect on vibration remove, mirage due to conversion in air, wide angle of camera lens, and compensation of misalignment angle, are discussed. The experimental results showed that our measurement technique is suitable for evaluation of high-temperature, large-scale infrastructure.

Determination of displacement distributions in bolted steel tension elements using digital image techniques

Digital imaging methods have found a great interest in various engineering fields to study stress-deformation characteristics of materials. Recent enhancements in visual instrumentation with the availability of cost-effective hardware and software products make the digital imaging techniques a viable tool to replace direct strain or displacement measurement methods in engineering applications. In this study, deformation characteristics of bolted steel connections are investigated by calculating in-plane displacement distributions using digital image correlation method (DIC). Validation of the method is presented by comparing the strains measured in standard tension specimens using electrical resistance strain gages and the DIC method. Finite element analysis of the connection specimen is also performed to compare the in-plane displacement distributions calculated from both methods. Results from the validation process indicate that the strains obtained from the DIC method compare well with the results of strain gages. The findings also indicate that the displacement distributions calculated from the finite element method may differ from those of the DIC method in terms of distribution pattern, and the location and magnitude of the extreme values of displacements. It is suggested that the proposed method can be used to determine the in-plane displacement distributions for the bolted connections, hence to evaluate their deformation characteristics under loading.

A Bending-Gradient model for thick plates, Part II: Closed-form solutions for cylindrical bending of laminates

In the first part ( Lebée and Sab, 2010a) of this two-part paper we have presented a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff–Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called Bending-Gradient plate theory is an extension to arbitrarily layered plates of the Reissner–Mindlin plate theory which appears as a special case when the plate is homogeneous. Moreover, we demonstrated that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner–Mindlin model. In this paper, the Bending-Gradient theory is applied to laminated plates and its predictions are compared to those of Reissner–Mindlin theory and to full 3D ( Pagano, 1969) exact solutions. The main conclusion is that the Bending-Gradient gives good predictions of deflection, shear stress distributions and in-plane displacement distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/ h goes to infinity.

A study on the digital moiré technique with circular and radial gratings

A digital moiré technique with circular and radial gratings is proposed to measure in-plane displacement and strain distributions of soft materials in a polar coordinate system. By introducing the coordinate-transform technique, the concentric circles and radial lines in the Cartesian coordinate system are converted to cross gratings in a polar coordinate system. Then, the digital moiré fringes are generated using logical and filter operations on the one-bit binary images. Finally, the in-plane displacement distributions are obtained by means of the four step phase-shifting procedure. Also, the measurement principles and the basic procedures of the new digital moiré technique are explained in detail. An application for rubber sheet with a single-edge crack is studied, and the experimental results show that the presented method is feasible.

3908 位相シフト画像の同時撮影によるモアレ干渉縞の位相解析(S17-2 光学的手法,S17 実験力学における最近の展開)

This paper describes a new simultaneous phase stepping method for moire interferometry. Conventional phase-stepping methods are widely accepted but the measured data are vulnerable to disturbances such as air fluctuations, temperature changes, vibrations and so on during the interferogram recording. To overcome this, a unique camera having a micro-retarder array on a CCD is used for observing interferograms. The specific structure makes it possible to calculate polarization states. Using this camera in a moire interferometer, the distribution of in-plane displacements is obtainable from a single image.

A higher order finite element theory for buckling and vibration analysis of initially stressed composite sandwich plates

A simple isoparametric assumed strain finite element formulation incorporating a third-order polynomial displacement model for the buckling and vibration analysis of initially stressed composite sandwich laminates is presented. The displacement model involves a nonlinear distribution of in-plane displacements through the plate thickness; the theory does not require shear correction coefficients. A nine-noded quadratic Lagrangian two-dimensional element is used with three displacements, two rotations of the normals about the plate midplane, and two warps of the normals. Full integration is carried out to evaluate various terms in the energy formulation. A consistent mass matrix is employed to preserve the total kinetic energy of the system. The accuracy of the present formulation is verified with the existing results in the literature. Numerical results are presented for the stability and free vibration of initially stressed composite sandwich plates.

OS01W0200 Sub-micron displacement measurement using a digital image correlation method

A high-precise measurement method using a digital image correlation method combined with a fine adjustment stage is proposed for the purpose of the thermal deformation measurement of small parts like the electronic devices. An object mounted on the fine adjustment stage is made to move parallel to the in-plane of the object surface at the displacement magnitude of the sub-micron and the images of the object surface are taken by a CCD camera. Calibration curves of the horizontal or vertical displacement vs. transfer pixel of the every pixel point of image are made by using the obtained images. In the case of disregarding out-of-plane deformation, it is possible to carry out the high-precise in-plane displacement distribution measurement using the calibration curves. As an application, the deformation of a driving electronic package is measured using this system.

A higher order shear deformable finite element for homogeneous plates

A C 0 finite element is introduced for the analysis of thick plates with transverse shear and normal strain and nonlinear in-plane displacement distribution with respect to the plate thickness. The warping theory proposed by Hassis [1] is used for the equations governing the plate deformation. The analytical solutions of the plate deformation theory were compared with other higher-order theories and found to be predicting the thick plate behavior reasonably well. Based on this higher order shear deformation theory, an eight-node finite element is introduced for thick plates, and a computer program is developed. The warping functions used in the formulations presented simpler equations than the other higher order homogeneous models. The proposed element incorporates bending–stretching coupling via the additional terms introduced in the displacement field. Some example problems are solved and the results are compared with the exact and other mathematical solutions available in the literature. For comparison, both stress and displacement results are investigated. The results of the proposed element are found to be in good agreement with the literature.

Minute displacement and strain analysis using lensless Fourier transformed holographic interferometry

An optical system for lensless Fourier transformed holographic interferometry is constructed to enable the measurement of minute displacements from nanometers to micrometers scale and to obtain corresponding strain distributions using a CCD camera with poor spatial resolution. Since a Fourier spectrum of an object beam is recorded on a hologram in this technique, the image reconstruction is easily performed with a single pass of 2-D fast Fourier transformation. Then, the map of the phase difference over the whole field is obtained by comparing two images before and after deformation. A suitable and effective unwrapping process is, however, inevitably required since the phase difference distribution is wrapped from − π to π in this technique. For phase unwrapping, the maximum spanning tree method is adopted here, which seeks a spanning tree that maximizes overall edge weights given by the cross amplitude. In-plane and out-of-plane displacements are obtained separately from the phase difference distributions at one's request. Moreover, in-plain strain is easily calculated from the in-plane displacement distribution.

A refined analysis of vibration and damping for anisotropic laminates in cylindrical bending

Abstract The finite element method based on transverse incompressibility and layer-wise linear distribution of in-plane displacements is applied to the dynamic problem of cross-ply and general laminates with off-axis layers in cylindrical bending. Specific damping capacity of each mode is obtained by the modal strain energy method. The numerical results for [0/90]slaminates show that through-the-thickness variation of in-plane displacements has a great influence on the damping of composite plates, especially on the damping of thick ones. Also, numerical examples are computed for an anisotropic laminates having stacking sequences such as [±θ]sand [θ/θ + 90]s, to investigate the effect of fiber orientation on fundamental frequencies and specific damping capacities. The present results show that the dynamic behaviors, especially specific damping capacity, of thick laminates are very different from those of thin laminates as the fiber orientation changes.

Bending solution of high-order refined shear deformation theory for rectangular composite plates

A new high-order refined shear deformation theory based on Reissner's mixed variational principle in conjunction with the state-space concept is used to determine the deflections and stresses for rectangular cross-ply composite plates. A zig-zag shaped function and Legendre polynomials are introduced to approximate the in-plane displacement distributions across the plate thickness. Numerical results are presented with different edge conditions, aspect ratios, lamination schemes and loadings. A comparison with the exact solutions obtained by Pagano and the results by Khdeir indicates that the present theory accurately estimates the in-plane responses.

Thermal buckling analysis of antisymmetric angle-ply laminates based on a higher-order displacement field

A higher-order deformation theory, which accounts for transverse shear and transverse normal strains, is derived for the thermal buckling analysis of antisymmetric angle-ply laminates that are simply supported and subject to a uniform temperature rise. The theory includes six dependent variables but can account for cubic distribution of in-plane displacements and parabolic distribution of transverse normal displacement. By using the present higher-order displacement field and the three-dimensional Hooke's law, exact-closed form solutions of the thermal buckling temperature are obtained. For purposes of comparison, numerical values of buckling temperature based on the first-order shear deformation theory and Reddy's higher-order shear deformation theory are also calculated, by reducing the present higher-order deformation theory. The results show that surprising discrepancies exist among the present theory and the other two theories, which indicates the importance of incorporating the effect of transverse normal strain in the thermal buckling analysis of antisymmetric angle-ply laminates.

A high-order laminated plate theory with improved in-plane responses

In order to improve the accuracy of the in-plane responses of the shear deformable laminated composite plate theories, a new high-order laminated plate theory was developed based upon Reissner’s new mixed variational principle [Int. J. Num. Meth. Eng.20, 1366 (1984)]. To this end, a zig-zag shaped C0 function and Legendre polynomials were introduced into the approximate in-plane displacement distributions across the plate thickness. The accuracy of the present theory was examined by applying it to the cylindrical bending problem of laminated plates which had been solved exactly by Pagano [J. Comp. Mat.3, 398 (1969)]. A comparison with the exact solutions obtained for several symmetric and asymmetric cross-ply laminates indicates that the present theory accurately estimates in-plane responses, even for small span-to-thickness ratios.