Shear flexible curved spline beam element for static analysis

doi:10.1016/S0168-874X(99)00013-X

Finite Elements in Analysis and Design

Volume 32, Issue 3, June 1999, Pages 181-202

https://doi.org/10.1016/S0168-874X(99)00013-X Get rights and content

Abstract

In this paper, an efficient curved cubic B-spline beam element is developed based on field consistency principle, for the static analysis. The formulation is general in the sense that it includes anisotropy, transverse shear deformation, in-plane and rotary inertia effects. The element is based on laminated beam theory, which satisfies the interface stress and displacement continuity, and has a vanishing shear stress on the top and bottom surfaces of the beam. The lack of consistency in the shear and membrane strain-field interpolations in their constrained physical limits causes poor convergence and unacceptable results due to locking. Hence, numerical experimentation is conducted to check these deficiencies with a series of assumed shear/membrane strain functions, redistributed in a field consistent manner. The performance of the element is assessed by studying the static behavior of a variety of problems ranging from straight beam to circular ring.

Introduction

The application of spline functions in the analysis of problems concerning structural mechanics has emerged to be an exciting area of research in the recent past. Due to their piecewise form, smoothness, capacity to handle local phenomena and higher-order continuity, spline functions offer distinct advantages such as computational efficiency, flexibility to model different boundary conditions, good accuracy and convergence characteristics, and versatility, etc. Among the spline functions, the functions based on B-spline basis, which can have different order of polynomial, is more in usage for the structural analysis.

The study of static and dynamic behavior of various structural elements, using different methods considering B-spline functions, have recently been carried out by many researchers. Some of the important contributions are cited here. Cheung et al. [1] and Cheung and Fan [2], have studied the static problems by employing spline finite strip method whereas finite element technique adopting spline functions has been used in the work of Shik [3] and Gupta et al. [4]. The dynamic characteristics of beams/plates and shells have been analyzed using spline finite point method by Zhou and Li [5]. Furthermore, finite element procedure using spline functions has been attempted for vibration study by Leung and Au [6], and Fan and Luah [7]. It may be observed from the existing literature that most of the available works applying various methods incorporating spline functions for structural analysis are based on the classical theory. It is further apparent from the available research works that the development of spline function based curved beam element has not been attempted, although several works are reported about straight beam, plate, and shell elements having functions based on spline bases.

It is a well-known fact that the classical theory is not suitable for analyzing either thick isotropic structures or even thin composite laminate cases, and therefore, it is more appropriate to analyze such structures by including the effect of shear deformation. However, the application of spline function in conjunction with shear deformation theory has been sparsely treated in the literature [8], [9], [10], [11], [12]. Spline collocation procedure has been adopted in Refs. [8], [9] to solve static problems whereas the vibration characteristics of straight beams and plates have been studied using spline functions in conjunction with Rayleigh–Ritz approach in Refs. [10], [11], [12]. It has been brought out from these studies that these methods yield results of good accuracy for moderately thick beam, plates and shells, but that accuracy of the results deteriorates significantly for thin structures due to shear locking phenomenon. The occurrence of shear locking phenomenon with respect to problems dealing with shear deformation theory in conjunction with B-spline functions has been eliminated by choosing different order of spline functions for the constrained field variables [10], by constructing B-spline displacement field based on the inspection of the Timoshenko beam mode functions [12], and by introducing modified shear modulii [8], [9]. The shear locking behavior in the finite element analysis of beams, plates and shells has been studied by many authors [13], [14], [15], [16], [17] in the context of elements based on conventional polynomial functions for the field variables. The more versatile approach, among the available techniques for alleviating locking such as reduced/selective integration scheme and assumed strain fields, etc., is the field-consistent formulation [16], [17]. It involves systematically eliminating spurious constraints causing shear/membrane locking in shear flexible finite elements. With such an approach, the order of integration required is freed and therefore, exact numerical integration schemes can be employed to evaluate all the strain energy terms. The performance of the elements based on such technique has been proved to be excellent for both thick and thin situations. An attempt is made here to develop a field-consistent shear-flexible curved beam element, adopting spline basis functions for the static analysis.

In this paper, we examine the cubic B-spline element from the point of view of field consistency to find the optimal assumed membrane/shear strain functions that will eliminate locking phenomena. The performance of the element is tested for static analysis of beams considering a number of problems. The results are compared, wherever possible, with the available analytical solutions.

Section snippets

Formulation based on laminated beam theory

A laminated composite beam, having radius of curvature R, is considered with the coordinates x along the axis of the beam and z along the thickness direction, respectively. The displacements in kth layer, u^k and w at point ( $x, z$ ) from the median surface are expressed as functions of mid-plane displacement u₀ and w and independent rotation θ of the normal in xz plane, as $u^{k} (x, z)=u_{0} (x) (1+z/R)−zw_{,x} (x)+[f (z)+g^{k} (z)] (w_{,x} (x)+θ(x)), w(x, z)=w(x).$ The functions $f (z)$ and g^k(z) are defined as $f (z)=t/ π sin (π$

Curved cubic B-spline beam element

A beam element is assumed to be having q equal sections. The spline function adopted to represent the three field variables u₀, w and θ is the cubic B-spline of equal section length (h), and is given as (Fig. 1) $u_{0} = ∑ i=−1 q+1 α_{i} ϕ^{0}_{i}, w= ∑ i=−1 q+1 β_{i} ϕ^{0}_{i}, θ= ∑ i=−1 q+1 γ_{i} ϕ^{0}_{i},$ in which each local cubic B-spline ϕ⁰_i has non-zero values over four consecutive sections with the section-knot x=x_i as the centre, and is defined as $ϕ^{0}_{i} = 1 6h^{3} 0, x<x_{i−2}, (x−x_{i−2})^{3}, x_{i−2} ⩽x⩽x_{i−1}, h^{3} +3h^{2} (x−x_{i−1})+3h(x−x_{i−1})^{2} −3(x−x_{i−1})^{3}, x_{i−1} ⩽x⩽x_{i}, h^{3}$

Numerical experiments

Numerical computation is carried out for the rank of the element and it has three proper zero values, and thus, does not produce any spurious mode. Further, for the critical evaluation of the present formulation, a series of test examples is considered to check the convergence properties and locking behavior. The element variations chosen for the study are the different functions for the redistributed strain fields through the field consistent strategy.

Conclusions

The shear-membrane locking phenomena in the shear flexible curved element based on cubic B-spline functions for the displacement fields have been analyzed using the field-consistency approach. The field-consistent redistribution of membrane and shear strain fields at the lowest level yields the accurate results for both fairly thick and extremely thin beams by using the full integration scheme for the evaluation of all the strain energy terms. The capabilities and effectiveness of the element

References (20)

A. Gupta et al.
Cubic B-spline for finite element analysis of axisymmetric shells
Comput. Struct.
(1991)
H.B. Zhou et al.
Free vibration analysis of sandwich plates with laminated faces using spline finite point method
Comput. Struct.
(1996)
A.Y.T. Leung et al.
Spline finite elements for beam and plate
Comput. Struct.
(1990)
S.C. Fan et al.
Free vibration analysis of arbitrary thin shell structures by using spline finite element
J. Sound Vib.
(1995)
I. Patlashenko et al.
Two-dimensional spline collocation method for nonlinear analysis of laminated panels
Comput. Struct.
(1995)
I. Patlashenko et al.
Cubic B-spline collocation method for nonlinear static analysis of panels under mechanical and thermal loadings
Comput. Struct.
(1993)
D.S. Malkus et al.
Mixed finite element methods-reduced and selective integration techniques: a unification of concepts
Comput. Methods Appl. Mech. Eng.
(1978)
G. Prathap et al.
Field-consistency analysis of the isoparametric eight-noded plate bending element
Comput. Struct.
(1988)
Y.K. Cheung, S.C. Fan, C.Q. Wu, Spline finite strip in structural analysis, Proceedings of the International...
Y.K. Cheung, S.C. Fan, Static analysis of right box girder bridges by finite strip method, Proc. Inst. Civ. Eng. Part 2...

There are more references available in the full text version of this article.

Cited by (23)

Nonlinear flutter of 2D variable stiffness curvilinear fibers composite laminates by a higher-order shear flexible beam theory with Poisson's effect
2022, Composite Structures
Citation Excerpt :
Such analysis is very useful for the design of flight vehicle structural panels made of variable stiffness curvilinear fibre based composite laminate exposed to supersonic flow. Large amplitude panel flutter phenomenon of laminated variable stiffness composite 2D panels reinforced with curvilinear fibres subjected to supersonic flow is focussed in this research work by adopting a sine function based shear flexible beam theory developed by Touratier [49], and Ganapathi et al. [50] and a C1 continuous beam finite element formulation. Aerodynamic pressure derived through first-order piston theory approximation is coupled with the proposed structural model.
In this work, the nonlinear supersonic panel flutter characteristics of two-dimensional variable stiffness curvilinear fibres based laminated composite panels are studied using a higher-order shear flexible theory represented by sine function coupled with first-order approximation leading to quasi-aerodynamic theory. The structural formation takes care of geometric nonlinearity with von Karman’s assumptions. The beam constitutive equation is modified for the laminated beam with general lay-up by accounting for Poisson’s effect. The nonlinear dynamic equilibrium equations developed by Lagrangian equations of motion are solved using finite element approach in conjunction with the direct iterative solution procedure. For limit cycle oscillation, critical dynamic pressure is predicted iteratively through eigenvalue analysis, thereby identifying the first coalescence of vibrational modes. Also, the flutter behavior of two-dimensional panel under static differential pressure is investigated considering nonlinear static equilibrium position of panel obtained by Newton-Raphson’s iterative approach and then followed by modes coalescence approach. These solution procedures are tested against the results in literature. A thorough numerical investigation is done to show the effect of the curvilinear fiber path orientation, limited cycle amplitude, static differential pressure, panel thickness, panel end condition flexibilities and thermal environment on the nonlinear supersonic panel flutter of two-dimensional variable stiffness laminated panels.
Variable stiffness composite laminated beams - nonlinear free flexural vibration behavior using a sinusoidal based shear flexible structural theory accounting for Poisson's effect
2022, International Journal of Non-Linear Mechanics
In the present work, formulation for the nonlinear free vibrational behavior of laminated variable stiffness composite beams with curvilinear fibers is made incorporating sinusoidal function for representing the transverse shear flexibility, geometrical nonlinearity, and accounting for Poisson’s effect through the constitutive equation for analyzing beams with general composite lay-up. Applying Hamilton’s principle combining with a finite element approach based on three-noded C¹ beam element, the governing equations are formed through matrices. The solutions for the developed governing equations are evaluated iteratively by introducing eigenvalue analysis and the results are viewed through the frequency–amplitude relationship. A large number of design parameters like curvilinear fiber angles in a layer, number of layers, lay-up sequence, thickness ratio, etc. is assumed to visualize the nonlinear free vibration characteristics of VSCL beams. Also, for the practical situation, the influence of thermal environment and restraint elastically against beam ends rotation, to accommodate other than classical boundary conditions, hybrid beam (consisting of constant and variable stiffness layers) on the non-linear free vibrational behavior is presented.
Nonlinear supersonic flutter study of porous 2D curved panels including graphene platelets reinforcement effect using trigonometric shear deformable finite element
2020, International Journal of Non-Linear Mechanics
The nonlinear panel flutter behaviour of two-dimensional porous curved panel reinforced by graphene platelets exposed to a supersonic flow is investigated. A curved beam element developed based on the trigonometric shear deformation theory is employed. The formulation integrates the geometric nonlinearity with von Karman’s approximation. The effort to model the fluid–structure interaction is reduced by implementing the first-order form of piston theory aerodynamics to describe the flow and accounting for the influence of static aerodynamic load due to the inherent geometric curvature of the panel. The nonlinear governing equations are formulated adopting the Lagrangian formulation. The panel deflection under the static aerodynamic load is evaluated using the Newton–Raphson iteration method. The flutter behaviour is analysed with reference to the large deflection equilibrium state through an eigenvalue analysis and by tracing the complex eigenvalues and identifying the first coalescence of any two vibratory modes. The flutter dynamic pressure is also predicted iteratively using the eigenvalue approach for the selected range of limit cycle amplitudes. The influence of static aerodynamic load and vibration amplitude on the flutter characteristics is brought out for both isotropic and graphene reinforced composite panels with different boundary conditions. The pre-flutter static deflection shape of the panel is also examined. The material parameters such as porosity level in the metal foam and the graphene platelet content are assessed on the nonlinear flutter features of 2D panels.
Supersonic flutter study of porous 2D curved panels reinforced with graphene platelets using an accurate shear deformable finite element procedure
2020, Composite Structures
The flutter behaviour of two-dimensional porous curved panels reinforced by graphene platelets exposed to supersonic flow on one side of the panels is investigated using the trigonometric shear deformation theory that satisfies stress free condition on the upper/lower surface of the panels. This structural model exhibits the thickness stretch effect thereby changing the transverse displacement. The effort to model the fluid-structure interaction is reduced by implementing the first-order approximation of piston theory aerodynamics to describe the flow. The solutions are found by introducing finite element methodology using a curved beam model. The critical flutter boundaries were predicted through the complex eigenvalue solution approach for the governing equations formulated adopting the Lagrangian formulation. Detailed numerical experimentation is made to show the effectiveness of the structural models, the influence of depth and length of curved panel, and panel edge conditions on the flutter boundaries of panels. Also, the material parameters such as porosity level, graphene platelet weight content, through-thickness distributions of nano-fillers and pores, size of nano-fillers are assessed on the flutter characteristics of 2D panels.
Nonlinear bending of porous curved beams reinforced by functionally graded nanocomposite graphene platelets applying an efficient shear flexible finite element approach
2020, International Journal of Non-Linear Mechanics
Citation Excerpt :
Before carrying out the detailed analyses, the behaviour of the model was examined by correlating the present model results with those of available in the open literature. The field-consistency based shear flexible element proposed here is assessed against the classical solution for extremely thin beam cases [38] in Table 1 and it shows very good agreement. Table 2 highlights the present results against the Navier solutions available for various theories [39] dealing with the bending of curved isotropic beams and they match very well.
In the present study, the nonlinear flexural bending of thick and thin porous curved composite beams reinforced and functionally graded by graphene platelets is carried out using a three-noded C¹ continuous curved beam finite element developed introducing an efficient shear deformation theory based on trigonometric function. The nonlinearity through the strain–displacement relationship by introducing von Karman’s assumptions is considered. The nonlinear equilibrium equations resulting from minimum potential energy principle are numerically solved based on the direct iteration technique. The bending nonlinear features through the load–deflection relationship are presented by selecting various design parameters like long and short beams, shallow and deep curved cases, support conditions, the variation of graphene platelets in the metal foam and the existence of porosity. This investigation reveals that the level of nonlinearity gets noticeably affected by the depth coupled with the curved beam slenderness ratio.
3D thermomechanical buckling analysis of perforated annular sector plates with multiaxial material heterogeneities based on curved B-spline elements
2018, Composite Structures
In the present paper, thermomechanical buckling of perforated functionally graded annular sector plates under uniform temperature rises and radial, circumferential or biaxial mechanical loads are investigated. Furthermore, effects of the circular cutouts on the buckling strength and deformation pattern are studied. Influence of the heterogeneity of material of the FGM sector in radial, circumferential, and transverse directions on the thermomechanical buckling is investigated for plates without or with one or two holes. Governing equations are derived based on the 3D energy-based theory of elasticity and buckling occurrence is detected by Treftz instability criterion. A novel curved 3D B-splined C²-continuous element is proposed to trace the inter-element variations of the displacements and stresses and model the geometric discontinuities (cutouts). A proper algorithm is also proposed to relate events of the original cylindrical coordinates to those of the natural coordinates and vice versa. Instead of using the common von Karman assumptions, the most general form of the strain tensor in curvilinear coordinates is used. Finally, effects of the sector dimensions, size and orientation of the cutout, heterogeneity direction, direction of the mechanical loads and the combination of the thermal and mechanical loads on the buckling loads and mode shapes are investigated.

View all citing articles on Scopus

View full text

Shear flexible curved spline beam element for static analysis

Abstract

Introduction

Section snippets

Formulation based on laminated beam theory

Curved cubic B-spline beam element

Numerical experiments

Conclusions

Comput. Struct.

Comput. Struct.

Comput. Struct.

J. Sound Vib.

Comput. Struct.

Comput. Struct.

Comput. Methods Appl. Mech. Eng.

Comput. Struct.