Deflection Assumption Research Articles

Geometrically nonlinear analysis of plates and shells by a cell-based smoothed CS-MITC18+ flat shell element with drilling degrees of freedom

A development of a 3-node triangular flat shell element to analyze geometric nonlinearity of plate and shell structures is presented in this study. The proposed flat shell element has 6 degrees of freedom per node, including the drilling rotational ones, by using the Allman-type approximations for the in-plane displacements and the bending displacement approximations enriched by a cubic shape function at the bubble node located at the element's centroid. The geometrically nonlinear behaviors are described by the von Karman's large deflection assumption. The membrane and bending strains of the presented flat shell elements are averaged over sub-triangular elements based on the cell-based smoothed (CS) technique. To attenuate the shear locking phenomenon, the transverse shear strains are re-interpolated following the mixed interpolation of tensorial components (MITC) technique designed for the 3-node triangular degenerated shell elements enriched by a cubic bubble function (MITC3+). Combined with the Newton-Raphson iterations and load increments predicted by the arc-length method, the suggested 3-node triangular flat shell elements, namely the CS-MITC18+ element, can analyze moderately geometrical nonlinearity, including the snap-thought and snap-back behaviors, of various plates and shells with the different shapes, thickness, and boundaries. The numerical investigations show that the load-displacement curves provided by the CS-MITC18+ flat shell elements well agree with other references.

Nonlinear buckling and postbuckling behaviors of porous FG-GPLRC cylindrical shells with stiffeners subjected to external pressure

Buckling and postbuckling behaviors of porous functionally graded graphene platelets-reinforced composite (porous FG-GPLRC) cylindrical shells with stiffeners subjected to external pressures are presented in this paper. Three distribution types of porosity in the shells are considered. The smeared technique for stiffeners is employed to model the mechanical behaviors of the stiffened shells. The mechanical formulations are established by the thin shell theory considering large deflection assumption, and the Ritz method is applied for three deflection amplitudes. The postbuckling formula of the pressure-deflection and the explicit critical buckling pressures can be achieved. The numerical investigations indicate the outstanding effects of stiffeners, porosity distribution, porosity coefficient, and graphene platelet (GPL) mass fraction on the nonlinear buckling responses of the stiffened shells.

Large Deflection Effects on the Energy Release Rate and Mode Partitioning of the Single Cantilever Beam Sandwich Debond Configuration

Abstract This paper investigates the effects of large deflections on the energy release rate and mode partitioning of face/core debonds for the single cantilever beam sandwich composite testing configuration, which is loaded with an applied shear force and/or a bending moment. Studies in this topic have been done by employing geometrically linear theories (either Euler–Bernoulli or Timoshenko beam theory). This assumes that the deflection at the tip of the loaded debonded part is small, which is not always the case. To address this effect, we employ the elastica theory, which is a nonlinear theory, for the debonded part. An elastic foundation analysis and the linear Euler Bernoulli theory are employed for the “joined” section where a series of springs exists between the face and the substrate (core and bottom face). A closed form expression for the energy release rate is derived by the use of J-integral. Another closed-form expression for the energy release rate is derived from the energy released by a differential spring as the debond propagates. Furthermore, a mode partitioning angle is defined based on the displacement field solution. Results for a range of core materials are in very good agreement with the corresponding ones from a finite element analysis. The results show that large deflection effects reduce the energy release rate but do not have a noteworthy effect on the mode partitioning. A small deflection assumption can significantly overestimate the energy release rate for relatively large applied loads and/or relatively long debonds.

Analytical prediction of tensile stress of photomask protective membranes based on measured frequency spectrum

The quality of photomask protective membranes is generally quantified in terms of its internal tensile stress. However, the tensile stress is not easily measured directly using experimental methods. Hence, the present research derives an theoretical model based on large deflection assumption for predicting the frequency spectrum of a pellicle. Using this model, the tensile stress of any membrane can be inversely derived by comparing the predicted frequency spectrum obtained using different tensile stress estimates with that determined experimentally using confocal microscopy and fast Fourier transform. The feasibility of the proposed approach is demonstrated by comparing the calculated frequencies of the main vibration modes of three commercial pellicles with the experimental results. The validated model is then used to investigate the effects of the membrane initial displacement, membrane width, and membrane thickness on the frequency response of a typical pellicle membrane.

Modified couple stress-based nonlinear static bending and transient responses of size-dependent sandwich microplates with graphene nanocomposite and porous layers

This paper investigates the nonlinear static bending and dynamic transient responses of the rectangular and circular sandwich microplates composed of functionally graded graphene nanocomposite (GNC) face sheets and a porous polymeric core layer based on the modified couple stress theory (MCST) within the isogeometric analysis (IGA) framework. The four-variable higher-order shear deformation refined plate theory (RPT) and the von Kármán large deflection assumption are synthetically employed to describe the kinematics of the microplate. The nonlinear governing equations of motion are derived from the Hamilton’s principle. The nonlinear bending problem is solved by the Newton–Raphson technique under a displacement criterion strategy, while the dynamic responses under transient loads are obtained by the Newmark integration scheme in conjunction with Newton–Raphson iterative procedure. Comparisons are carried out to test the accuracy of iteration procedure in present formulation and the reliability in capturing small scale effect of present model. Obtained outcomes reveal that raising the material length scale (MLS) parameter and volume fraction index of graphene nanofillers both leads to reductions in nonlinear static deflections, dynamic amplitudes of deflection and periods of motion. The examples for porosity coefficient show that embedding pores in core layer leads to slight changes in terms of static and dynamic responses and an improvement in weight reduction of the sandwich microplate.

A Concise and Geometrically Exact Planar Beam Model for Arbitrarily Large Elastic Deformation Dynamics.

The potential of large elastic deformations in control applications, e.g., robotic manipulation, is not yet fully exploited, especially in dynamic contexts. Mainly because essential geometrically exact continuum models are necessary to express these arbitrarily large deformation dynamics, they typically result in a set of nonlinear, coupled, partial differential equations that are unsuited for control applications. Due to this lack of appropriate models, current approaches that try to exploit elastic properties are limited to either small deflection assumptions or quasistatic considerations only. To promote further exploration of this exciting research field of large elastic deflection control, we propose a geometrically exact, but yet concise a beam model for a planar, shear-, and torsion-free case without elongation. The model is derived by reducing the general geometrically exact the 3D Simo–Reissner beam model to this special case, where the assumption of inextensibility allows expressing the couple of planar Cartesian parameters in terms of the curve tangent angle of the beam center line alone. We further elaborate on how the necessary coupling between position-related boundary conditions (i.e., clamped and hinged ends) and the tangent angle parametrization of the beam model can be incorporated in a finite element method formulation and verify all derived expressions by comparison to analytic initial value solutions and an energy analysis of a dynamic simulation result. The presented beam model opens the possibility of designing online feedback control structures for accessing the full potential that elasticity in planar beam dynamics has to offer.

Isogeometric analysis for size-dependent nonlinear free vibration of graphene platelet reinforced laminated annular sector microplates

The proposed work fills a gap of study on size-dependent large amplitude free vibration of functionally graded graphene platelets-reinforced composite (FG-GPLRC) annular sector microplates. Based on a four-variable higher-order shear deformation plate theory, the von Kármán large deflection assumption and the modified couple stress theory (MCST), the governing equations for the discrete nonlinear free vibration of the multilayer FG-GPLRC annular sector microplate without considering the external loads are established by the principle of virtual work. The NURBS-based isogeometric analysis (IGA) in conjunction with a displacement control strategy are utilized synthetically to acquire linear natural frequencies and nonlinear-to-linear natural frequency ratios for the microplates numerically. Comparisons and convergence study are carried out to verify accuracy and correctness of IGA and iteration method in present formulation. Detailed parametric studies are performed to present an insight into the influences of distribution pattern and weight fraction of the graphene platelets, sector angle and material length scale parameter on the behaviors of linear natural frequencies and large amplitude responses of the microplates under different boundary conditions.

Isogeometric analysis for postbuckling of sandwich cylindrical shell panels with graphene platelet reinforced functionally graded porous core

The proposed work investigates for the first time the postbuckling behavior of sandwich cylindrical shell panels with two metallic face layers and a graphene platelet (GPL) reinforced functionally graded porous (FGP) core subjected to central point loads, uniform and non-uniform pressure loadings. Based on a general higher-order shear deformation shell theory and the von Kármán large deflection assumption, the discrete equilibrium equations of the sandwich cylindrical panels are established by the principle of virtual work. The NURBS (non-uniform rational B-spline) based isogeometric analysis (IGA) in conjunction with the modified arc-length method of Crisfield are utilized synthetically to acquire nonlinear load–deflection responses for the panels numerically and both snap-through and snap-back types of instability are captured. Several benchmarks are carried out to verify accuracy and reliability of IGA and arc-length method in present formulation. Parametric studies are performed to demonstrate the influences of weight fraction of the graphene platelet, porosity distribution, porosity coefficient and thickness of the sandwich core on the nonlinear load–deflection responses of the sandwich cylindrical panels. Some complex load–deflection curves of the FGP-GPL cylindrical panels may be useful for future references.

Response investigation of viscoelastic cylindrical shells with geometrical nonlinearity effect under moving pressure: An analytical approach

In this research, an analytical method is presented to investigate the nonlinear response of viscoelastic cylindrical shells with the finite length under moving internal pressure by moderately large deflection assumption. The viscoelastic behavior is considered as elastic in the bulk and standard linear solid in the shear. The strain-displacement relations are defined using nonlinear von Karman relations. By employing Hamilton’s principle, the equations of motion are extracted based on the classical shell theory. These equations which are a system of nonlinear coupled partial differential equations, are solved by the method of multiple scale of the perturbation technique and the response due to moving pressure and the critical velocity are determined. Moreover, the effects of different geometrical and viscoelastic parameters on the results are investigated. The results are compared with the finite element method and the reported results in some literatures.

Analyses of thermal buckling and secondary instability of post-buckled S-FGM plates with porosities based on a meshfree method

The present research deals with thermal buckling and postbuckling of sigmoid functionally graded material (S-FGM) plates with porosities resting on elastic foundation based on the popular third-order shear deformation theory and the well-known von Kármán large deflection assumption. Special attention is paid to scrutinize the secondary instability that occurs during postbuckling regime. In addition, the transitions of deflection shapes of the plates under various boundary conditions between primary and secondary postbuckling paths are investigated. A displacement control strategy and a meshfree radial point interpolation method (RPIM) which employs a radial basis function without any adaptive parameters are utilized synthetically to carry out thermal buckling loads and postbuckling paths of the plates numerically. Numerical comparisons are performed to verify the convergence of the modified RPIM and the accuracy of iteration method in postbuckling path analysis. The influences of functionally graded index, porosity coefficient and parameters of elastic foundation to behaviors of the thermal buckling and especially the secondary instability occurs during postbuckling regime are investigated. It is found that the secondary instability is sensitive to the boundary conditions.

The Strength of Egg Trays under Compression: A Numerical and Experimental Study.

This work concerns the analysis of egg packages subjected to compression. Experimental investigations were carried out to determine the curves of compression and maximum loads. To compare packages accessible on the market, several different shapes of egg packages were tested after being conditioned in air with a relative humidity of 50%. Several paper structures in stock were compressed. By validating the experiment results, numerical computations based on the finite element method (FEM) were executed. The estimations of a numerical model were performed with the use of the perfect plasticity of paper and with the assumption of large strains and deflections. Our own two structures of egg packaging were taken into account: basic and modified. The material of the packages was composed of 90% recovered paper and 10% coconut fibres. This paper involved the numerical modelling of such complex packaging. Moreover, our research showed that introducing several features into the structures of the packaging can improve the stiffness and raise the maximum load. Thanks to the application of ribs and grooves, the strength ratio and compression stiffness, in comparison to the basic tray, increased by approximately 23.4% and 36%, respectively. Moreover, the obtained indexes of modified trays were higher than the majority of the studied market trays.

The Effect of Large Deflections of Joints on Foldable Miniature Robot Dynamics

In miniature robotics applications, compliant mechanisms are widely used because of their scalability. In addition, compliant mechanism architecture is compatible with the manufacturing methods used to fabricate small scale robots, such as “foldable robotics”, where the size and the materials used allow much larger deflections. In this paper, the kinematics of compliant mechanisms used in miniature foldable robots are investigated with the assumption of nonlinear large deflections that occur at the flexure joints. The solution of the large beam deflection is acquired using elliptic integrals and is verified with finite element analysis and experiments on a simple small foldable leg linkage. The large deflection model takes joint strain energies into account and yields accurate estimations for load capacity of the mechanism as well as the necessary input torque for actuation of the leg. Therefore, the model presented can be used to estimate the load capacity of a miniature robot, as well as to select appropriate actuators. The work is also extended to estimate the compliant leg kinematics and rigid body dynamics of a foldable robot. The robot’s large deflection simulation results are compared with experiments and a simplified rigid-link pin-joint kinematic model. Our results demonstrate the modeling accuracy of the two approaches and can be used by foldable robotics community when deciding on the strategy to choose for modeling their robots.

Geometrically nonlinear analysis of functionally graded shells using an edge-based smoothed MITC3 (ES-MITC3) finite elements

An edge-based smoothed finite element method (ES-FEM) combined with the mixed interpolation of tensorial components technique (MITC) for triangular elements, named as ES-MITC3, was recently proposed to enhance the accuracy of the original MITC3 for analysis of shells. In this study, the ES-MITC3 is extended to the geometrically nonlinear analysis of functionally graded shells. In the ES-MITC3, the stiffness matrices are obtained using the strain smoothing technique over the smoothing domains that formed by two adjacent MITC3 triangular shell elements sharing an edge. The material properties of functionally graded (FG) shells are assumed to vary through the thickness direction by a power rule distribution of volume fractions of the constituents. The nonlinear finite element formulation based on the first-order shear deformation theory with the von Karman’s large deflection assumption is used to describe the large deformations of the FG shells. Several numerical examples are given to demonstrate the performance of the present approach in comparison with other existing methods.

Design, analysis and validation of the bridge-type displacement amplification mechanism with circular-axis leaf-type flexure hinges for micro-grasping system

For stable and flexible manipulation, realizing parallel grasping and large output stroke simultaneously has become a key issue in the researches on micro-grasping system. The typical solution is to use compliant orthogonal displacement amplification mechanism (such as bridge-type mechanism), in which the flexure hinges are generally axial load-dominated, whereas the deflection of traditional straight-axis flexure hinges is not related to the axial load under small deflection assumption. In this study, a bridge-type mechanism with circular-axis leaf-type (CALT) flexure hinges is designed, analyzed and validated. Through modeling the generalized displacements of CALT flexure hinge, the positive direction of its axis in the bridge-type mechanism is designed. The precise models of load relations for the CALT flexure hinges in the bridge-type DAM are established, and the multi-objective parametric optimization is further given. Small deflection-based static FEA results verify the generalized displacements’ models of CALT flexure hinge as well as the parametric optimization result. Compared with the corresponding bridge-type mechanism with the largest output displacement in the traditional researches, the output displacement of optimal result is improved effectively. For the design validation, the optimal design result is further applied to construct a piezoelectric-driven microgripper, which grasps parallelly and enlarges the output stroke simultaneously.

Nonlinear analyses of FGM plates in bending by using a modified radial point interpolation mesh-free method

A modified mesh-free radial point interpolation method (RPIM) which employs a new radial basis function capable of building the shape functions without any supporting fixing parameters is developed for the nonlinear bending analyses of functionally graded material (FGM) plates. The shape functions constructed by the RPIM possess Kronecker delta function property and thus special schemes for imposing boundary conditions are not needed. The nonlinear bending formulations of FGM plates are based on a higher-order shear deformation plate theory considering the von Kármán large deflection assumption. The nonlinear equations are solved by the modified Newton–Raphson iterative technique. The computed results by the modified RPIM are compared with the other numerical solutions reported in the literature to verify the effectiveness and accuracy of the proposed mesh-free method. Detailed parametric studies are then carried out to scrutinize the effects that the volume fraction, plate length-to-thickness ratio, boundary condition and initial deflection have on the nonlinear flexural responses of FGM plates. Results illustrate that the modified mesh-free RPIM can effectively predict the nonlinear bending behavior of FGM plates.

Nonlinear thermal buckling analyses of functionally graded plates by a mesh-free radial point interpolation method

This study intends to analyze nonlinear buckling behavior of functionally graded (FG) plates under thermal loading by a mesh-free method. The buckling formulation is derived based on the higher-order shear deformation plate theory in which the von Kármán large deflection assumption is employed. An improved mesh-free radial point interpolation method (RPIM) which incorporates the normalized radial basis function capable of building the shape functions without any fitting parameters is presented and utilized to scrutinize the buckling responses. The nonlinear equations are solved by the modified Newton–Raphson iterative technique. Verification of the improved RPIM is implemented by simulating several numerical examples available in the literature and comparing the outcomes with the analytical results. Detailed parametric studies demonstrate that the improved mesh-free RPIM can effectively predict the thermal buckling responses of FG plates, and the volume fraction, plate length-to-thickness ratio, aspect ratio, boundary condition have considerable effects on the critical buckling temperatures of FG plates subjected to various types of temperature variations through the thickness.

Vibration of thermally post-buckled hybrid laminates with two non-uniformly distributed fibers

This study presents the first vibration analysis of thermally post-buckled hybrid laminates with non-uniformly distributed graphite and E-glass fibers in a single lamina. A 54 degree-of-freedom high order triangular plate element is developed based on the Von Karman large deflection assumption. The formulation of the location dependent linear, nonlinear stiffness and mass matrices due to non-homogeneous material properties, geometrical stiffness matrix due to thermal effects and thermal moment vector due to temperature gradient is derived. The effects of hybrid-fiber distribution on the natural frequencies and mode shapes of thermally post-buckled laminates are discussed. The numerical results reveal that the redistribution of two fibers can considerably decrease the postbuckling deflections of the hybrid laminates and significantly modify the natural frequencies. The stiffening effect of fiber redistribution is more obvious for the laminate with a higher volume fraction index and clamped edges. Special buckling and vibration mode shapes are observed, along with multiple vibration mode shifting.

Free vibration behavior of viscoelastic annular plates using first order shear deformation theory

In this paper, an analytical procedure based on the perturbation technique is presented to study the free vibrations of annular viscoelastic plates by considering the first order shear deformation theory as the displacement field. The viscoelastic properties obey the standard linear solid model. The equations of motion are extracted for small deflection assumption using the Hamilton

Eigensensitivity analysis of rotating clamped uniform beams with the asymptotic numerical method

In this paper, free vibrations of a rotating clamped Euler–Bernoulli beams with uniform cross section are studied using continuation method, namely asymptotic numerical method. The governing equations of motion are derived using Lagrange׳s method. The kinetic and strain energy expression are derived from Rayleigh–Ritz method using a set of hybrid variables and based on a linear deflection assumption. The derived equations are transformed in two eigenvalue problems, where the first is a linear gyroscopic eigenvalue problem and presents the coupled lagging and stretch motions through gyroscopic terms. While the second is standard eigenvalue problem and corresponds to the flapping motion. Those two eigenvalue problems are transformed into two functionals treated by continuation method, the Asymptotic Numerical Method. New method proposed for the solution of the linear gyroscopic system based on an augmented system, which transforms the original problem to a standard form with real symmetric matrices. By using some techniques to resolve these singular problems by the continuation method, evolution curves of the natural frequencies against dimensionless angular velocity are determined. At high angular velocity, some singular points, due to the linear elastic assumption, are computed. Numerical tests of convergence are conducted and the obtained results are compared to the exact values. Results obtained by continuation are compared to those computed with discrete eigenvalue problem.

Effect of linear and non-linear blade modelling techniques on simulated fatigue and extreme loads using Bladed

There is a trend in the wind industry towards ever larger and more flexible turbine blades. Blade tip deflections in modern blades now commonly exceed 10% of blade length. Historically, the dynamic response of wind turbine blades has been analysed using linear models of blade deflection which include the assumption of small deflections. For modern flexible blades, this assumption is becoming less valid. In order to continue to simulate dynamic turbine performance accurately, routine use of non-linear models of blade deflection may be required. This can be achieved by representing the blade as a connected series of individual flexible linear bodies - referred to in this paper as the multi-part approach. In this paper, Bladed is used to compare load predictions using single-part and multi-part blade models for several turbines. The study examines the impact on fatigue and extreme loads and blade deflection through reduced sets of load calculations based on IEC 61400-1 ed. 3. Damage equivalent load changes of up to 16% and extreme load changes of up to 29% are observed at some turbine load locations. It is found that there is no general pattern in the loading differences observed between single-part and multi-part blade models. Rather, changes in fatigue and extreme loads with a multi-part blade model depend on the characteristics of the individual turbine and blade. Key underlying causes of damage equivalent load change are identified as differences in edgewise- torsional coupling between the multi-part and single-part models, and increased edgewise rotor mode damping in the multi-part model. Similarly, a causal link is identified between torsional blade dynamics and changes in ultimate load results.