Strain Energy Of Plate Research Articles

Localization of Vibration Weak Position of Composites Based on Weighted Modal Strain Energy Summation

In this paper, two typical examples are used to illustrate the weak position of aircraft structure in the process of vibration. Through the modal analysis of the typical composite plate and I-shaped beam, the first 20-order modal strain energy of the plate is extracted, which is difficult to locate the weak spot due to the highly scattered location of the higher modal strain energy. The modal participation factor is introduced as the weight factor of the summation of the modal strain energy. The modal participation factor is large, the weighting factor is large, and the high modal strain energy of the composite plate moves diagonally in the 45° direction of the composite plate and the high strain energy region is consistent with the previous modes of the plate. This is the result of the weak in-plane shear stiffness of the composite panel, which shows the effectiveness of the mode weighted summation method. The I-shaped composite beam uses the modal strain energy summation of the weight factor, and the higher modal strain energy is concentrated on the middle part of the beam and at 1/4 and 3/4 of it. Therefore, the weak part of the vibration can be clearly identified. The higher modal strain energy is extracted by the method proposed to this paper, which can be used as a reference to structural design and dynamic on-line monitoring.

Size-Dependent Buckling and Post-Buckling Analysis of the Functionally Graded Thin Plate Al-Cu Material Based on a Modified Couple Stress Theory.

Size-dependent functionally graded material thin plate buckling and post-buckling problems are considered using the framework of the MCST (Modified Couple Stress Theory). Based on modified couple stress theory and power law, the post-buckling deflection and critical buckling load of simply supported functionally graded material thin plate are derived using Hamilton’s minimum potential energy principle. The analysis compares the simulation results of linear buckling and nonlinear buckling. Innovatively, a power-law distribution with scale effects is considered. The influences of scale effect parameters l and power-law index parameters k on buckling displacement, load, and strain energy of plates have been investigated. In this article, it is found that the critical buckling displacement, critical buckling load, and buckling strain energy increase with increases in the power-law index parameters k. The membrane energy decreases as the power-law index parameter increases. If the upper and lower layers are swapped, the opposite result is obtained. In comparison, the scale effect parameter is more influential than the power-law exponent. The critical buckling displacement in the x-direction is not affected by scale effects. The critical buckling load, the membrane energy, and buckling strain energy increase as the scale effect parameter increases. Scale effects increase material stiffness compared with traditional theory, and the power-law index parameters affect FGM properties such as elastic modulus, Poisson’s ratio, density, etc. Both scale effects parameters and power-law index parameters have important effects on the mechanical behavior of materials.

Optimal Design and Biomechanical Analysis of a Biomimetic Lightweight Design Plate for Distal Tibial Fractures: A Finite Element Analysis

The treatment of fractures of the distal tibia can be problematic due to the insubstantial soft-tissue covering this part of the anatomy. This study investigates a novel strategy for minimally invasive plate osteosynthesis of distal tibia fractures called bionic lightweight design plating. Following the structure of the animal trabecular bone, we utilized topological mathematical methods to redesign the material layout of the internal fixation device to fulfill the desired lightweight design within given boundary conditions. The results showed that this method can maintain the same stability of the construct as the original plate after a reduction in the original volume by 30%, and the differences in strain energy of plates and maximum node displacement of constructs between the constructs [RP construct vs. LP construct] were not statistically significant (p > 0.05). In the safety assessment of the constructs, the peak stress of plates between constructs was found to not be statistically significantly different under a doubled physiological load (p > 0.05). The average stress of the plates’ elements exceeding the allowable stress was analyzed, and no statistically significant differences were found between the two constructs under axial compression stress conditions (p > 0.05). The average stress of the plates’ elements in the redesigned plating construct under torsional stress conditions was 3.08% less than that of the locked plating construct (p < 0.05). Under the double physiological load condition, 89% of the elements of the plate in the redesigned plating construct and 85% of the elements of the plate in the locked plating construct were lower than the maximum safe stress of the plate, which was 410 MPa (secondary allowable stresses). That reminds us the topology optimization offer a possible way to improve the capacity of soft tissue protection while ensuring the safety of the RP construct by reducing the volume of the implants.

Bending analysis of skew ribbed plates with a meshfree method

Based on the first - order shear deformation theory (FSDT), a meshless method for solving the free vibration problem of the skew ribbed plates is proposed. The plates and ribs are discretized with a series of points to obtain a meshless model of the ribbed plate by simulating the ribs with beams. The FSDT and the moving least - squares approximation are used to establish the displacement field. The displacement fields of the stiffener can be expressed in terms of the mid-surface displacement of the plate by imposing displacement compatible conditions between the plate and the stiffener. Then the stiffness equation of the entire skew ribbed plate can be derived by superimposing the strain energy of plate and stiffener. And the boundary condition is introduced by the full transformation method. Several examples are calculated, and the results given by the proposed method are compared with those from other researchers or ABAQUS. The results show that the method can effectively analyze the bending problem of the skew ribbed plate, and it avoids redistribution of plate nodes when rib position changes.

Vortex dynamics and flow-induced vibrations arising from a vortex ring passing tangentially over a flexible plate

The extraction of energy from vortical structures advecting through an ambient environment is a topic of interest due to the potential to power miniature in situ sensors and monitors. This work investigates the vortex dynamics and flow-induced vibrations of a flexible plate arising from a vortex ring passing tangentially over it. Experimental measurements of the flow field and plate dynamics are performed in tandem with a coupled potential flow/Kirchhoff-Love plate model in order to (i) elucidate the physics of the vortex-plate interactions in the specified orientation and relate the energy exchange between the ring and the plate to the attendant vortex dynamics; (ii) validate the potential flow model and provide any needed corrections to account for the simplifying assumptions; and (iii) provide empirical data for estimating energy harvesting capabilities in the specified orientation. The plate loading arises as a result of an initial down-wash, followed quickly by a region of reduced pressure as the vortex core passes over the plate. The fundamental physics of the interaction is discussed, identifying three regimes. When the centerline of the vortex ring is positioned greater than approximately 2 vortex ring radii away from the plate it can be considered to be in the far-field, and the resulting vibrations are well predicted through potential flow, once the plate dynamics are corrected for edge effects arising from a finite plate width. As the offset distance of the vortex ring is decreased, diffusion of induced vorticity on the plate into the flow field significantly alters the fluid dynamics, pressure loading, and the resultant plate dynamics, and dramatically increases the strain energy in comparison with the potential flow model predictions. A first-order correction to the potential flow model is proposed to account for the finite plate width, while empirical correlations are presented for the plate strain energy in cases where ring/induced vorticity interactions are significant.

Dynamics of an inverted flexible plate in a uniform flow

The dynamics of an inverted flexible plate with a free leading-edge and a fixed trailing-edge in a uniform flow has been studied numerically by an immersed boundary-lattice Boltzmann method for the fluid flow and a finite element method for the plate deformation. Mechanisms underlying the dynamics of the fluid-plate system are elucidated systematically. A series of distinct states of the plate deformation and motion are identified and can be described as straight, flapping, deflected, deflected-flapping, and asymmetric-flapping states. Which state to occur depends mainly on the bending stiffness and aspect ratio of the plate. The forces exerted on the plate and the elastic strain energy of the plate are analyzed. It is found that the flapping state can improve the conversion of fluid kinetic energy to elastic strain energy. In addition, the effects of the mass ratio of the plate and the fluid, the Reynolds number, and the angle of attack of the uniform flow on the dynamics and the elastic strain energy of flexible plate are also investigated in detail. The vortical structures around the plate are given to discuss the connection of the evolution of vortices with the plate deformation and motion. The results obtained in this study provide physical insight into the understanding of the mechanisms on the dynamics of the fluid-plate system.

Static and free vibration analyses of stiffened folded plates using a cell-based smoothed discrete shear gap method (CS-FEM-DSG3)

Recently, a cell-based smoothed discrete shear gap method (CS-FEM-DSG3) based on the first-order shear deformation theory (FSDT) was proposed for static and free vibration analyses of Mindlin plates. The CS-FEM-DSG3 uses three-node triangular elements that can be easily generated automatically for arbitrary complicated geometric domains. In this paper, the CS-FEM-DSG3 is further extended for static and free vibration analyses of stiffened folded plates, by combining the original CS-FEM-DSG3 with an Allman's plane stress triangular element and with a stiffener modeled by Timoshenko beam element. The model of a stiffened folded plate is formed by (1) regarding the plate and the stiffener separately in which the stiffeners can be placed anywhere on the plate and need not be placed along the mesh lines, (2) imposing displacement compatible conditions between the plate and the stiffener so that displacement fields of the stiffener can be expressed in terms of the mid-surface displacement of the plate, (3) superimposing the strain energy of plate and stiffener, and (4) using a transformation of degrees of freedom from a local system to a global system. The accuracy and reliability of the proposed method are verified by comparing its numerical solutions with those of other available numerical methods.

SIMP-Based Dynamic Topological Optimum depending on Maximum Eigenfrequency for Reinforcement of Concrete Beams

Computer aided topology optimization design is a relatively new but rapidly expanding area of structural mechanics. Topology optimization design is used in an increasing rate by for example building and bridge engineering as well as the car, machine and aerospace industries. The reason for this is that it often achieves great savings and design improvements by solving the basic engineering problem of distributing a limited amount of material in a design space, in which a certain objective function has to be optimized. In static problems a common objective is to minimize the compliance such as strain energy and in dynamic problems the first natural eigenfrequency is often maximized in order to evaluate the stiffest structures. The goal of this study is to get a wide use for structural designs in the topology optimization. Therefore with respect to SIMP dynamic topology optimization problems are treated in comparisons with static problems in this research. For the topology optimization problems implemented algorithms can be the optimality criteria method and the method of moving asymptotes (MMA). Numerical applications topologically maximizing the first natural eigenfrequency or minimizing strain energy of plates verify the generality and wide use of topology optimization with respect to structural designs.

Stability Analysis of Stitched Composite Plate System with Delamination Under Hygrothermal Pressure

In this study, a very recent stitched composite structure made of fiber reinforced polymers is analyzed. This composite structure is modeled by two specially orthotropic adhesively bonded, delaminated and stitched plates under in-plane compressive and shear loads and internal hygro-thermal pressure acting on delamination surfaces of plates which are considered as identical simply supported plates. While the adhesive layer between plate layers is modeled by mechanical linear normal and shear springs, stitches are modeled by discrete bilinear normal springs for the first time in the literature. In the stitch modeling, it is assumed that they can carry only normal loads because they are very thin and also initial imperfections are included with the parameter of looseness because of the fact that the stitching yarns are slack initially. The mathematical model of the problem is solved by Rayleigh-Ritz method. After strain energies of plates, the adhesive layer and stitches and energy potentials for external loads such as compressive normal loads, shear loads and internal hygro-thermal pressure are expressed, sine series as shape functions for plate vertical deflections satisfying simply supported boundary conditions are applied. Then, minimizing them gives deflections for the given loading. The stability condition is obtained as an eigenvalue problem to give critical buckling loads and corresponding mode shapes as well. Results are presented as plots of deformed shapes of the system and tables of parametric studies showing effects of changing parameters on stitch stresses and displacements, critical buckling loads and corresponding mode shapes.

Buckling and free vibration analyses of stiffened plates using the FSDT mesh-free method

This paper presents a mesh-free Galerkin method for the free vibration and stability analyses of stiffened plates via the first-order shear deformable theory (FSDT). The model of a stiffened plate is formed by (1) regarding the plate and the stiffener separately, (2) imposing displacement compatible conditions between the plate and the stiffener so that displacement fields of the stiffener can be expressed in terms of the mid-surface displacement of the plate, and (3) superimposing the strain energy of plate and stiffener. Because there are no meshes used in this method, the stiffeners can be placed anywhere on the plate and need not be placed along the mesh lines. Several numerical examples are computed by this method to show its accuracy and convergence. The present results demonstrate good agreement with the existing solutions given by other researchers and the ANSYS. Influences of support size and order of the complete basis functions on the numerical accuracy are also investigated.

Optimum Placement and Control of Active Constrained Layer Damping using Modal Strain Energy Approach

The Active Constrained Layer Damping (ACLD) treatment has been used successfully for controlling the vibration of various flexible structures. It provides an effective means for augmenting the simplicity and reliability of passive damping with the low weight and high efficiency of active controls to attain high damping characteristics over broad frequency bands. In this paper, optimal placement strategies of ACLD patches are devised using the modal strain energy (MSE) method. These strategies aim at minimizing the total weight of the damping treatments while satisfying constraints imposed on the modal damping ratios. A finite element model is developed to determine the modal strain energies of plates treated with ACLD. The treatment is then applied to the elements that have highest MSE in order to target specific modes of vibrations. Numerical examples are presented to demonstrate the utility of the devised optimization technique as an effective tool for selecting the optimal locations of the ACLD treatment to achieve desired damping characteristics over a broad frequency band.

Pointwise energy release rate in delaminated plates

A laminated plate theory suitable for analyzing delaminations has been derived. The theory is used to study the interaction between the top and bottom sublaminates in the intact region of a delaminated plate. Expressions are derived for the jump in force and moment resultants that occur across the delamination front. Using Irwin's crack closure integral, a simple expression for pointwise strain energy release rate G along the delamination front has been derived. The expression suggests that the G at any point on the delamination front is the difference between the plate strain energy densities behind and ahead of the delamination front. An estimate of error in computing G using plate theories is obtained by comparing the J integral obtained using exact stress fields and plate stresses. The procedure for computing G is first verified by applying it to double cantilever beam specimens (DCB) and elliptical delaminations in isotropic plates for which solutions are available or can be computed. Then the method is illustrated for a stitched graphite/epoxy DCB specimen and also for elliptical delaminations in 0-deg graphite/epoxy plates. The results demonstrate the usefulness of the present method in analyzing delaminated coupons and structures.

On the strain energy of laminated composite plates

A strain energy function is obtained for nonhomogeneous. laminated, composite plates for the case when each lamina exhibits monoclinic material symmetry about its middle surface. The starting point is the three-dimensional strain energy based on geometrically nonlinear elasticity theory. The variational-asymptotical method is used to decompose the nonlinear three-dimensional problem into two separate problems: (1) a linear, through-the-thickness. one-dimensional analysis to obtain appropriate plate elastic constants and relations between plate deformation variables and three-dimensional results: and (2) a nonlinear, two-dimensional analysis to analyse the plate deformation. Closed-form analytical expressions are derived for the plate elastic constants as well as the displacement and strain distributions through the thickness of the plate. Even with this generality, no more variables are involved than in Rcissner Mindlin plate theory. Also, in spite of the simple form for the plate strain energy, there are no restrictions on the magnitudes of displacement and rotation measures. Two approximations are obtained in the through-thc-thickness analysis, the tirst being equivalent to classical laminated plate theory, and the second incorporating shear deformation etfects. The first approximation is asymptotically correct for plates of the form considered. The second approximation is asymptotically correct for plates with certain additional material restrictions. In applying the method, one first solves the through-thc-thickness problem and then uses the resulting elastic constants to pose the nonlinear plate problem. After solving the nonlinear problem, one substitutes these results back into the linear three-dimensional relations for displacement and strain throughout the plate.

Induced strain actuation of isotropic and anisotropic plates

The development and experimental verification of the induced strain actuation of plate components of an intelligent structure is presented. Equations relating the actuation strains, created by induced strain actuators, to the strains induced in the actuator/substrate system are derived for isotropic and anisotropic plates. Plate strain energy relations are also developed. Several exact solutions are found for simple actuator/substrate systems, and a general procedure for solving the strain energy equations with a Rayleigh-Ritz procedure is formulated. Approximate Ritz solutions lead to both an understanding of the system design parameters, and to detailed models of cantilever plate systems. Simple test articles were used to verify the accuracy of the basic induced strain actuator/substrate system models, and cantilever plate test articles were built and tested to verify the ability of the models to predict the strains induced in systems with extensive stiffness coupling and complicated boundary conditions.

Discussion of “ Contribution of Gaussian Curvature to Strain Energy of Plates ” by Cheng Zhu and Frank L. DiMaggio (July, 1989, Vol. 115, No. 7)

Discussion of “ <i>Contribution of Gaussian Curvature to Strain Energy of Plates</i> ” by Cheng Zhu and Frank L. DiMaggio (July, 1989, Vol. 115, No. 7)

Discussion of “ Contribution of Gaussian Curvature to Strain Energy of Plates ” by Cheng Zhu and Frank L. DiMaggio (July, 1989, Vol. 115, No. 7)

Discussion of “ <i>Contribution of Gaussian Curvature to Strain Energy of Plates</i> ” by Cheng Zhu and Frank L. DiMaggio (July, 1989, Vol. 115, No. 7)

Discontinuous Dependency of Plate Strain Energy on Boundary Contour

This note shows that the strain energy of a plate suffers a finite jump when the boundary contour of the plate is subject to a change, however small, from a smooth curve to a piecewise linear curve, if either the displacement or the slope, both of which are prescribed, is homogeneous at every point along the plate boundary. The proof is based on an observation that the plate strain energy depends linearly on Poisson's ratio when the lateral load, the boundary displacement and slope are specified. It also makes use of a previous result that the Gaussian curvature term in plate strain energy is a discontinuous functional of the boundary contour, and extends the theorem to that of the plate strain energy itself. This note suggests that this singular behavior may result from a discontinuous dependency of the solution to plate equation on small changes of the domain of existence.

Contribution of Gaussian Curvature to Strain Energy of Plates

It is shown that when a plate with a smooth boundary, on every point of which either the displacement or slope is zero, there is, in general, a contribution to the flexural strain energy from the Gaussian curvature term. When the plate boundary is approximated by a polygon with a large, but finite, number of sides, it is shown that the contribution of the Gaussian curvature, converted to a line integral around the boundary, is zero both along the sides and on corners where the displacement gradient is continuous. The singuiar behavior is shown to be the result of discontinuities of the Gaussian curvature term as a functional of the boundary contour.

A 36 DOF Symmetrically Laminated Triangular Element with Shear Deformation and Rotatory Inertia

An approach is presented in the formulation of a 36 d.o.f. symmetrically laminated composite triangular plate element including the enect of shear deformation for free vibra tion analysis. The strain energy of plates can be expressed as the sum of the flexural and shear strain energies. The total displacement is expressed as the sum of the displacement due to bending and that due to shear deformation. Thus only the displacement due to bend ing appear in the flexural strain energy and the displacement due to shear deformation ap pear in the shear strain energy. Numerical results for natural frequencies for a range of dif ferent isotropic, orthotropic and anisotropic plates with various thickness-to-length ratios are obtained and compared with various alternative solutions available in the literature to demonstrate the valid range of applicability of this approach for the free vibration of plates with the effect of shear deformation.

An approach to the inclusion of transverse shear deformation in finite element plate bending analysis

An expression for the strain energy of plates, which includes a term for transverse shear deformation and which is expressed in terms of the total transverse displacement and the displacement due to bending alone, is presented. Use of this expression in the construction of finite element representations requires independent descriptions of both these displacements. A triangular plate element is formulated in this manner, using the shape functions of the “discrete-Kirchoff” model. Numerical results for a range of finite element meshes and thickness-to-plate length ratios, for rectangular plates with different support and loading conditions, demonstrate the effectiveness of the proposed approach.