Derivatives Of Transverse Displacement Research Articles

Nonlinear aerodynamic analysis of functional graded plates using an HSDT-based isogeometric approach

In this paper, a nonlinear FGM plate analysis model is presented based on IGA, which suppresses shear locking and is applicable to plates with a wide range of thickness-to-length ratios. Furthermore, the nonlinear aerodynamic characteristics of FGM plate subjected to the combined effect of transverse load, in-plane load, and supersonic flow are investigated. HSDT is used to construct a displacement field suitable for nonlinear vibration analysis of FGM plates with a wide range of thickness-to-length ratios. For the problem that shear locking cannot be avoided entirely in the nonlinear analysis of plates, the first derivative of transverse displacement is adopted as the first term of HSDT, which weakens the effect of shear strain. Therefore, the shear locking is effectively restrained. The nonlinear FGM plate analysis model is derived using NURBS, which meets the C1 continuity requirement of HSDT. The computational efficiency, accuracy and the suppressing capacity to shear locking are verified in three aspects: static displacement, eigenfrequency, and nonlinear frequency ratio. The Bathe method is used to obtain the displacement response of an FGM plate subjected to supersonic flow, in-plane load, and transverse load. The dynamic characteristics of the FGM plate are studied using time history, bifurcation diagrams, PSD curves, phase diagrams, and Poincaré maps.

Continuum models for pantographic blocks with second gradient energies which are incomplete

We postulate a deformation energy for describing the mechanical behavior of so called pantographic blocks, that is bodies constituted by stacking of N layers of pantographic sheets. We remark that the pantographic effect is limited in the plane of pantographic sheets and therefore only the second derivatives of transverse displacements along the pantographic fibers appear in the chosen deformation energy. We use this novel energy to predict the behavior of pantographic blocks when subjected to : (i) compression and traction test, (ii) torsion, (iii) shear and (iv) bending. A linearization of the energy shows one floppy mode in addition to the rigid body motions, assuming perfect pivots, similar to pantographic sheets.

An equivalent single-layer theory for free vibration analysis of steel-concrete composite beams

An equivalent single-layer theory (EST) is put forward for analyzing free vibrations of steel-concrete composite beams (SCCB) based on a higher-order beam theory. In the EST, the effect of partial interaction between sub-beams and the transverse shear deformation are taken into account. After using the interlaminar shear force continuity condition and the shear stress free conditions at the top and bottom surface, the displacement function of the EST does not contain the first derivatives of transverse displacement. Therefore, the C0 interpolation functions are just demanded during its finite element implementation. Finally, the EST is validated by comparing the results of two simply-supported steel-concrete composite beams which are tested in laboratory and calculated by ANSYS software. Then, the influencing factors for free vibrations of SCCB are analyzed, such as, different boundary conditions, depth to span ratio, high-order shear terms, and interfacial shear connector stiffness.

Efficient 2D thermo-mechanical analysis of composites and sandwich laminates

ABSTRACTThe static behavior of composites and sandwich plates in thermo-mechanical environment is investigated by a two dimensional (2D) FE model. An efficient higher-order zig-zag theory (HOZT) considering actual through-thickness temperature profile and a least square error (LSE) method to accurately predict the inter-laminar stresses is implemented in this model. The in-plane displacement field is obtained by superposing a cubically varying global displacement field on a zig-zag displacement field having different slopes at each layer. This plate theory represents parabolic through thickness variation of transverse shear stresses, which satisfy the inter-laminar continuity at the layer interfaces and zero transverse shear stress conditions at the top and bottom of the plate. In the present 2D finite element (FE) model, the first derivatives of transverse displacement have been treated as independent variables to circumvent the problem of C1 continuity associated with the above plate theory (HOZT). The accurate through-thickness distribution of temperature is obtained by using a linear zig-zag thermal lamination theory proposed by the authors by using the thermal conduction properties of different constituent layers in the thickness direction. The LSE method is applied at the postprocessing stage to accurately calculate the inter-laminar stresses by the 3D equilibrium equations of the plate problem, after in-plane stresses are calculated. The proposed combined FE model (HOZT+LSE) is implemented to analyze the static behavior of laminated composites and sandwich plates subjected to thermo-mechanical loadings. Many new results are also presented that should be useful for the future reference.

Analytical analysis for vibration of longitudinally moving plate submerged in infinite liquid domain

The vibration of a longitudinally moving rectangular plate submersed in an infinite liquid domain is studied analytically with the Rayleigh-Ritz method. The liquid is assumed to be incompressible, inviscid, and irrotational, and the velocity potential is used to describe the fluid velocity in the whole liquid field. The classical thin plate theory is used to derive mechanical energies of the traveling plate. As derivative of transverse displacement with respect to time in the compatibility condition equation exists, an exponential function is introduced to depict the dynamic deformation of the moving plate. It is shown that this exponential function works well with the Rayleigh- Ritz method. A convergence study shows a quick convergence speed for the immersed moving plate. Furthermore, the parametric study is carried out to demonstrate the effect of system parameters including the moving speed, the plate location, the liquid depth, the plate-liquid ratio, and the boundary condition. Results show that the above system parameters have significant influence on the vibration characteristics of the immersed moving plate. To extend the study, the method of added virtual mass incremental (AVMI) factor is used. The results show good agreement with those from the Rayleigh-Ritz method.

Hygrothermal analysis of laminated composites using C0 FE model based on higher order zigzag theory

A C0 FE model developed based on an efficient higher order zigzag theory is used for hygrothermal analysis of laminated composite plates. The C0 FE model satisfies the inter-laminar shear stress continuity at the interfaces and zero transverse shear stress conditions at plate top and bottom. In this model the first derivatives of transverse displacement have been treated as independent variables to circumvent the problem of C1 continuity associated with the above plate theory. In the present theory the above mentioned C0 continuity of the present element is compensated in the stiffness matrix formulation by using penalty parameter approach. In order to avoid stress oscillations observed in the displacement based finite element, the stress field derived from temperature/moisture fields (initial strains) must be consistent with total strain field. Special steps are introduced by field consistent approach (e.g., sampling at gauss points) to compensate this problem. A nine noded C0 continuous isoparametric element is used in the proposed FE model. Comparison of present numerical results with other existing solutions shows that the proposed FE model is efficient, accurate and free of locking.

Hygrothermal analysis of laminated composite plates in terms of an improved C0-type global–local model

In this paper, an efficient improved C0-type global–local model (IGLM) considering transverse normal strain is proposed to study the hygrothermal response of thick cross-ply laminated composite plates. Based on the interlaminar continuity conditions of in-plane displacement and transverse shear stresses, layer-dependent variables could be reduced. Employing shear stress free condition at the upper and the lower surfaces, derivatives of transverse displacement are eliminated from the displacement field, so that C0 interpolation functions are only required for the finite element implementation. As a result, the number of variables is independent of the number of layers of the laminate. To assess the proposed model, the classical quadratic eight-node isoparametric element is presented, in which the interelement C0 continuity conditions are satisfied. Comparing the results from available three-dimensional elasticity theory and those computed from the ninth-order model, it is found that the simple C0 finite elements could produce promising deformations and stresses for thick cross-ply laminated composite plates with different boundary conditions under hygrothermal loadings.

Modeling of functionally graded sandwich shells accounting for variation in transverse displacement

ABSTRACTIn this study, a simple C0 isoparametric finite element formulation based on higher-order shear deformation theory is presented for static analysis of functionally graded material sandwich shells (FGMSS). To characterize the membrane-flexure behavior observed in a functionally graded shell, a displacement field involving higher-order terms in in-plane and transverse fields is considered. The proposed kinematics field incorporates for transverse normal deformation, transverse shear deformation, and nonlinear variation of the in-plane displacement field through the thickness to predict the overall response of the shell in an accurate sense. To develop the efficient C0 formulation, the derivatives of transverse displacement are treated as independent field variables (nodal unknowns). Voigt's rule of mixture is employed to ascertain the mechanical properties of each layer's constituents along the thickness direction. A wide range of numerical problems are solved assuming various parameters: side-thickness ratio, curvature-side ratio, and gradation parameter, and their interactions with regard to static analysis of FGMSS are discussed in brief. Deflection and stresses incorporating different thickness schemes of sandwich shells are presented in the form of figures. To validate the results, a functionally graded shell without sandwich arrangement is considered. Since no results are available on static analysis of FGMSS, the present 2D model based on the finite element method might be helpful in assessing the applicability of other analytical and numerical models in this area in the future.

C<sup>0</sup>-type Reddy's theory for composite beams using FEM under thermal loads

To analyze laminated composite and sandwich beams under temperature loads, a C0-type Reddy\'s beam theory considering transverse normal strain is proposed in this paper. Although transverse normal strain is taken into account, the number of unknowns is not increased. Moreover, the first derivatives of transverse displacement have been taken out from the in-plane displacement fields, so that the C0 interpolation functions are only required for the finite element implementation. Based on the proposed model, a three-node beam element is presented for analysis of thermal responses. Numerical results show that the proposed model can accurately and efficiently analyze the thermoelastic problems of laminated composites.

A C0 zig-zag model for the analysis of angle-ply composite thick plates

From a theoretical and practical viewpoint, the zig-zag theory is well adopted in the analysis of laminated composite structures. Nevertheless, for the available zig-zag models, artificial constraints in which the first derivatives of transverse displacement are replaced by the assumed variables have to be employed to avoid C1 interpolation functions in the finite element implementation. Such artificial constraints violate continuity conditions of interlaminar transverse stresses at interfaces. To avoid using artificial constraints, a C0-type zig-zag model is proposed in this paper. C0 interpolation functions are only required in the finite element formulation as first derivatives of transverse displacement have all been eliminated from the displacement field based on stress compatibility conditions between plies and on the top and bottom surfaces of the plate. Moreover, the number of variables involved in the proposed zig-zag model is less than that of the existing zig-zag models, yet accurate results are produced comparable to analytical solutions and three-dimensional finite element results. Effects of ply orientations, boundary conditions and length-to-thickness ratio on displacements and stresses of laminated composite plates have been studied.

Calculation of interlaminar shear stresses in laminated shallow shell panel using refined higher order shear deformation theory

In general, calculation of inter-laminar shear stresses in laminated shells using 2D finite element models involves cumbersome post processing techniques. This paper presents a simple and efficient method for accurate evaluation of transverse shear stresses in laminated composite shallow shells by using a displacement based C0 2D FE model derived from refined higher order shear deformation theory (RHSDT) and a least square error (LSE) method. The theory satisfies the inter-laminar shear stress continuity conditions at the layer interfaces and zero transverse shear stress conditions at the top and bottom of the plate. The effect of three curvature terms in the strain components of composite shells is also considered by following the Sander’s approximations. In order to overcome the problem of C1 continuity of transverse displacement encountered at the time of FE implementation of the present shell theory (RHSDT), the first derivatives of transverse displacement have been replaced by a independent C0 variables. The LSE method is applied at the post-processing stage, after in-plane stresses are calculated by using the present FE model based on RHSDT. Thus the proposed method is quite simple and efficient compared to the usual method of integrating the 3D equilibrium equations for calculation of transverse stresses in laminated composite shells. Accuracy of the method is demonstrated in the numerical examples through comparison of the present results with those obtained from different models based on refined higher order shear deformation theory (RHSDT), higher order shear deformation theory (HSDT), exact analytical and 3D elasticity solutions.

Efficient C0 Finite Element Formulation for Thermal Stress Analysis of Laminated Composite and Sandwich Plates

AbstractBy introducing thermal expansion coefficients in the transverse displacement field, this paper proposes an approach to improve the performance of the global-local higher-order theory for thermal stress analysis of laminated composite plates. Transverse normal deformation has been considered in the proposed model, whereas the additional displacement variables have not increased because thermal loads could be included in the generalized force vector. The proposed model a priori satisfies the continuity conditions of transverse shear stresses at interfaces, and the number of displacement variables involved in the present model is independent of the number of layers. The derivatives of transverse displacement have been eliminated from the displacement field so that the C0 interpolation functions are only required for the finite-element implementation. Based on the proposed model, a six-node triangular element is proposed for thermal expansion problems of laminated composite and sandwich plates. Numeri...

Indirect boundary force measurements in beam-like structures using a derivative estimator

This paper proposes a new method for the identification of boundary forces (shear force or bending moment) in a beam, based on displacement measurements. The problem is considered in terms of the determination of the boundary spatial derivatives of transverse displacements. By assuming the displacement fields to be approximated by Taylor expansions in a domain close to the boundaries, the spatial derivatives can be estimated using specific point-wise derivative estimators. This approach makes it possible to extract the derivatives using a weighted spatial integration of the displacement field. Following the theoretical description, numerical simulations made with exact and noisy data are used to determine the relationship between the size of the integration domain and the wavelength of the vibrations. The simulations also highlight the self-regularization of the technique. Experimental measurements demonstrate the feasibility and accuracy of the proposed method.

Buckling of soft-core sandwich plates with angle-ply face sheets by means of a C0 finite element formulation

In order to avoid using C1 interpolation functions in finite element implementation of the previous zig–zag theories, artificial constraints, in which the first derivatives of transverse displacement will be replaced by the assumed variables, are usually employed. However, such assumption will violate continuity conditions of transverse shear stresses at interfaces. Differing from previous work, this paper will propose a C0-type zig–zag theory for buckling analysis of laminated composite and sandwich plates with general configurations. The first derivatives of transverse displacement have been taken out from a displacement field of the proposed zig–zag theory. Thus, the C0 interpolation functions are only required in finite element implementations of the proposed model. Without use of any artificial constraints, an eight-node quadrilateral element based on the proposed model is presented by incorporating the terms associated with the geometric stiffness matrix. In order to verify performance of the proposed model, several buckling problems of sandwich plates with soft core have been analyzed. Numerical results show that the proposed model is able to predict accurately buckling loads of the soft-core sandwich plates with varying fiber orientations of face sheets.

Effect of interfacial imperfection on bending behavior of composites and sandwich laminates by an efficient C0 FE model

The bending behavior of composites and sandwich plates having imperfections at the layer interfaces is investigated by a refined higher order shear deformation plate theory (RHSDT) and a Least Square Error (LSE) method. In this theory, the in-plane displacement field is obtained by superposing a globally varying cubic displacement field on a zig-zag linearly varying displacement field. This plate theory represents parabolic through thickness variation of transverse shear stresses which satisfy the inter-laminar continuity condition at the layer interfaces and zero transverse shear stress condition at the top and bottom of the plate. In this plate model, the interfacial imperfection is represented by a liner spring-layer model. Finite element method is adopted and an efficient C0 continuous 2D finite element (FE) model is developed based on the above mentioned plate theory for the static analysis of composites and sandwich laminates having imperfections at the layer interfaces. In this model, the first derivatives of transverse displacement have been treated as independent variables to circumvent the problem of C1 continuity associated with the above plate theory (RHSDT). The LSE method is applied to the 3D equilibrium equations of the plate problem at the post-processing stage, after in-plane stresses are calculated by using the above FE model based on RHSDT. The proposed model is implemented to analyze the laminated composites and sandwich plates having interfacial imperfection. Many new results are also presented which should be useful for the future research.

Static, Vibration and Buckling Analysis of Skew Composite and Sandwich Plates Under Thermo Mechanical Loading

Static, vibration and buckling behavior of laminated composite and sandwich skew plates is studied using an efficient C<sup>0</sup> FE model developed based on refined higher order zigzag theory. The C<sup>0</sup> FE model satisfies the interlaminar shear stress continuity at the interfaces and zero transverse shear stress conditions at plate top and bottom. In this model, the first derivatives of transverse displacement have been treated as independent variables to overcome the problem of C<sup>1</sup> continuity associated with the plate theory. The C<sup>0</sup> continuity of the present element is compensated in the stiffness matrix formulation by adding a suitable term. In order to avoid stress oscillations observed in the displacement based finite element, the stress field derived from temperature is made consistent with the total strain field by using field consistent approach. Numerical results are presented for different static, vibration and buckling problems by applying the FE model under thermo mechanical loading, where a nine noded C<sup>0</sup> continuous isoparametric element is used. It is observed that there are very few results available in the literature on laminated composite and sandwich skew plates based on refined theories. As such many new results are also generated for future reference

Inter-laminar stresses in a laminated shallow shell panel

The evaluation of inter-laminar shear stresses in laminated shells using 2D finite element models involves cumbersome post-processing techniques. In this paper, a simple and efficient method has been proposed for accurate evaluation of transverse shear stresses in laminated composite shells by using a displacement-based C0 FE model derived from higher-order shear deformation theory (HSDT) and a least square error (LSE) method. In order to include the effect of three curvature terms in the strain components of composite shells, Sander’s approximations are followed. In this model, the first derivatives of transverse displacement have been treated as independent variables to overcome the problem of C1 continuity in the FE implementation associated with the present shell theory (HSDT). The LSE method is applied at the post-processing stage, after in-plane stresses are calculated by using the present FE model based on HSDT. Thus, the proposed method is quite simple compared to the usual method of integrating the 3D equilibrium equations for the calculation of transverse stresses in laminated composite shells. The accuracy of the method is demonstrated in the numerical examples by comparison of the present results with those obtained from different models based on HSDT, exact analytical and 3D elasticity solutions.

Free Vibration Analysis of Laminated Soft Core Sandwich Plates

Free vibration behavior of laminated soft core sandwich plates with stiff laminated face sheets is investigated using a new C0 finite element (FE) model based on higher order zigzag theory (HOZT) in this paper. The in-plane displacement variations are considered to be cubic for both the face sheets and the core, while the transverse displacement is assumed to vary quadratically within the core and remains constant in the faces beyond the core. The plate theory ensures a shear stress-free condition at the top and bottom surfaces of the plate. Thus, the plate theory has all of the features required for an accurate modeling of laminated sandwich plates. As very few elements based on this plate theory (HOZT) exist and they possess certain disadvantages, an attempt has been made to develop this new element. The nodal field variables are chosen in such a manner to overcome the problem of continuity requirement of the derivatives of transverse displacements, i.e., no need to impose any penalty stiffness in the formulation. A nine node C0 quadratic plate finite element is implemented to model the HOZT for the present analysis. A new C0 element has been utilized to study some interesting problems on free vibration analysis of laminated sandwich plates. Many new results are also presented which should be useful for future research.

Identification of Beam Boundary Conditions Using Displacement Derivatives Estimations

The purpose of the study is to illustrate the possibility of boundary conditions identification in case of beams. The problem is understood as a determination of various spatial derivatives of transverse displacements. Using the assumption that displacements can be approximated by Taylor expansions on a domain near its boundaries, the spatial derivatives are estimated using particular pointwise derivatives estimators. This approach enables the extraction of these quantities using a weighted spatial integration of the displacement field. Numerical simulation using exact and noisy data are shown in order to clarify the link between the size of the integration domain and the wavelengths of the vibrations.

Buckling analysis of laminated composite plates using an efficient C0 FE model

Buckling analysis of laminated composite plates is carried out by using an efficient C0 FE model developed based on higher order zigzag theory. In this model the first derivatives of transverse displacement have been treated as independent variables to overcome the problem of C1 continuity associated with the FE implementation of the plate theory. The C0 continuity of the present FE model is compensated in the stiffness matrix calculations by using penalty parameter approach. Numerical results and comparison with other existing solutions show that the present model is very efficient in predicting the buckling responses of laminated composites.