Geometrically nonlinear isogeometric analysis of laminated composite plates based on higher-order shear deformation theory

In this paper, we present an effectively numerical approach based on isogeometric analysis (IGA) and higher-order shear deformation theory (HSDT) for geometrically nonlinear analysis of laminated composite plates. The HSDT allows us to approximate displacement field that ensures by itself the realistic shear strain energy part without shear correction factors. IGA utilizing basis functions namely B-splines or non-uniform rational B-splines (NURBS) enables to satisfy easily the stringent continuity requirement of the HSDT model without any additional variables. The nonlinearity of the plates is formed in the total Lagrange approach based on the von-Karman strain assumptions. Numerous numerical validations for the isotropic, orthotropic, cross-ply and angle-ply laminated plates are provided to demonstrate the effectiveness of the proposed method

Weakly Enforced Boundary Conditions for the NURBS-Based Finite Cell Method

In this paper, we present a variationally consistent formulation for the weak enforcement of essential boundary conditions as an extension to the finite cell method, a fictitious domain method of higher order. The absence of boundary fitted elements in fictitious domain or immersed boundary methods significantly restricts a strong enforcement of essential boundary conditions to models where the boundary of the solution domain coincides with the embedding analysis domain. Penalty methods and Lagrange multiplier methods are adequate means to overcome this limitation but often suffer from various drawbacks with severe consequences for a stable and accurate solution of the governing system of equations. In this contribution, we follow the idea of NITSCHE [29] who developed a stable scheme for the solution of the Laplace problem taking weak boundary conditions into account. An extension to problems from linear elasticity shows an appropriate behavior with regard to numerical stability, accuracy and an adequate convergence behavior. NURBS are chosen as a high-order approximation basis to benefit from their smoothness and flexibility in the process of uniform model refinement

Numerical Analysis of Stress-Strain State of Orthotropic Plates in the Form of Arbitrary Convex Quadrangle

A numerical and analytical approach to solving problems of the stress-strain state of quadrangular orthotropic plates of complex shape has been proposed. Two-dimensional boundary value problem was solved using spline collocation and discrete orthogonalization methods after applying the appropriate domain transform. The influence of geometric shape of plate in different cases of boundary conditions on the displacement and stress fields is considered according to the refined theory. The results were compared with available data from other authors

Isogeometric analysis for functionally graded microplates based on modified couple stress theory

Analysis of static bending, free vibration and buckling behaviours of functionally graded microplates is investigated in this study. The main idea is to use the isogeometric analysis in associated with novel four-variable refined plate theory and quasi-3D theory. More importantly, the modified couple stress theory with only one material length scale parameter is employed to effectively capture the size-dependent effects within the microplates. Meanwhile, the quasi-3D theory which is constructed from a novel seventh-order shear deformation refined plate theory with four unknowns is able to consider both shear deformations and thickness stretching effect without requiring shear correction factors. The NURBS-based isogeometric analysis is integrated to exactly describe the geometry and approximately calculate the unknown fields with higher-order derivative and continuity requirements. The convergence and verification show the validity and efficiency of this proposed computational approach in comparison with those existing in the literature. It is further applied to study the static bending, free vibration and buckling responses of rectangular and circular functionally graded microplates with various types of boundary conditions. A number of investigations are also conducted to illustrate the effects of the material length scale, material index, and length-to-thickness ratios on the responses of the microplates.Comment: 57 pages, 14 figures, 18 table

Analysis of plates and reinforced concrete columns by cubic b-spline function

Applying spline functions to numerical structural analysis has been more common in recent years. The recent increase use of spline functions is mainly due to their excellent characteristics, such as sectionalized continuity, linear combination, flexibility, and easy use for various boundary conditions. The whole deformed shape of some structures or substructures can be described with one displacement function constructed by a series of spline functions. By doing this, the mesh generation and the huge computer memory space are no longer needed because only one single superelement can be used in the whole process. The choice of spline functions as displacement functions has many advantages that have been demonstrated by several researchers for a limited range of structures. More extensive research on using spline functions in structural analysis hereafter can be expected. This research work represents an effort in that direction. Based on cubic B-spline functions, this dissertation presents static and free vibration analysis of arbitrary quadrilateral flexural plates with various boundary conditions. Combination of cubic B-spline functions in two orthogonal directions constructs a superelement for the whole plate. The cubic B-spline displacement function has been formed to efficiently model the deflection shape and to yield more accurate results. A further step has been taken in the present research to apply the cubic B-spline function to a nonlinear problem. A numerical method is developed for the determination of complete load-deflection and moment-curvature relationships for slender reinforced concrete columns with arbitrary cross sections under combined biaxial flexure and axial load. Improvement of computer time and accuracy has been demonstrated obviously due to the application of cubic B-spline function and introduction of p-multiplier in the numerical formulation. Comparison of present analysis with analytical solutions, other numerical methods, and experimental results, appears to have a good agreement

Direct Method of Analysis of an Isotropic Rectangular Plate Using Characteristic Orthogonal Polynomials

This work evaluates the static analysis of an isotropic rectangular plate with various boundary conditions. The direct variation method according to Ritz is used to obtain the total potential energy of the plate by employing the static elastic theory of plate. The shape function of fourth order for the plate under uniformly distributed load for various boundary conditions were formulated using the characteristic orthogonal polynomials (COP), which is also known as product of orthogonal strips of the plate along the two axes. Analyses of results obtained were compared with exact results presented in the monograph of Timoshenko and Woinowsky – Krieger. The previous work adopted shape function in the form of trigonometric series to solve the fourth order governing differential equation of the plate. This work is helpful for obtaining the shape functions not only for simply supported plates but also for rectangular plates with various kinds of support conditions, where the exact method presents intricate and complex solution. Finally, comparison has been done for the coefficients of deflection and moments at various aspect ratios of the plate for different support conditions. The results obtained by the use of the COP in the classical Ritz method are in close agreement with the results obtained from exact solutions from classical method.http://dx.doi.org/10.4314/njt.v34i2.