Plate Bending Spline Finite Strip Models Using Mixed Formulation and Combined Spline Series

DOI: http://dx.doi.org/10.24018/ejers.2019.4.10.1489 42  Abstract—As Classical and Spline finite strip method based on stiffness and mixed variational formulation principle become important tool for continuum structural analysis, especially in the field of plate bending problem, a lot of researches has been focused on interpolation functions in order to improve the efficiency and increase the reliability of the method. The main objective of this paper is to introduce and propose a new spline interpolation function in the light of combination techniques of basic splines through introduction and brief review of previous studies in this field. This work which uses abbreviated form of augmented matrix proposed by authors published in previous work, reveals a very good results when compared with the analytical and published solutions of different plate bending problems.


I. INTRODUCTION
Since the spline finite strip method based on stiffness formulation was introduced in the early eighties, although it is capable of solving many engineering problems such as plate bending [1]- [3], vibration [4] [5], and buckling [6] [7], it is subjected to many improvements such as the mixed formulation technique based on variational principles, where the authors previous work leads to the availability of the explicit form of the Augmented Matrix of such formulation [8], consequently the branch of improvement of interpolation functions take the place.This paper is aimed at studying several B-spline modifications for their applicability in modeling of engineering problems which frequently requires the restraint of a number of interior nodes as well as describes the construction of combined splines (CB-spline) as interpolation function in order to overcome on some difficulties of conventional spline finite strip plate bending, and thus a new type of CB-spline interpolation is presented.This interpolation allows the user to introduce interior supports at any location in much the same way as in a standard finite element method.This option significantly increases the applicability of spline interpolation, moreover this work will pave the way to handle many other problems in the field of vibrations and buckling of plates as well as in shell analysis.The paper is Published on October 17, 2019.A. M. Ahmed is an Assistant Professor, Basic Science Department, College of Electronics Technology, Bani-walid, Libya, (e-mail: ENG.Abdarrhim@Gmail.com)S. Y. Barony was a Professor of Structural Engineering, with Al-Fateh University, Tripoli, Libya.He is now with the Department of Civil Eng.Department, School of Engineering., Libyan Academy of Graduate Studies, Tripoli, Libya.(e-mail: Saleh.barony@ academy.edu.ly).organized as follows: section 2 presents variational principle functional with the corresponding produced augmented matrix produced due it.Section 3 surveys the proposed mixed spline plate models.In section 4 a vector space concepts were shortly revised mathematically.Section 5 covers the important key points concerned with standard Bspline interpolation.Combination of basic splines (B-Splines) for interpolation is introduced in section 6.This paper end with implementation of the new proposed combined spline function through a series of numerical examples in section 7 and finally the main objectives were concluded and the versatility of the new approach is demonstrated.

II. VARIATIONAL PRINCIPLE AND MIXED FORMULATION FOR PLATE BENDING FINITE STRIP
Based on Kirchhoff plate theory, the Mixed Formulation for isotropic thin plate bending strip with homogenous boundary conditions according to Hellinger-Reissner functional [9]- [11]: is fully discussed in the authors previous work [8] and the augmented matrix was obtained as:

III. MIXED SPLINE STRIP MODELS FOR PLATE BENDING
From [8], the so-called basic matrix [  ] , ℐ,ℓ was given as:

A. Low Order Strip (LO2)
The augmented elemental matrix [  ] for LO2-plate bending strip was given as: and the sectional matrix [  ] , as:

B. High Order Strip (HO3)
The augmented elemental matrix for HO3-plate bending strip was given as: and the sectional augmented matrix as: The coefficients of these sub-matrices were given in [8].

IV. MATHEMATICAL CONCEPTS
A vector can be defined as anything we want it to be; an ordered n-tuple, a number, or even a function.The vectors that are elements in a special kind of set called a vector space.The set of those elements on which two operations called vector addition and scalar multiplication are defined, is said to be a vector space if they satisfy all the closure axioms listed for vector spaces.In the elementary vector courses, any vector  ⃗ ⃗ in 3D-space can be written as a linear combination of the well-known linearly independent standard basis vectors for the system of 3D-vectors that is:  ⃗ ⃗ =  1  +  2  +  3 , where the standard basis vectors are i = < 1, 0, 0 >, j = < 0, 1, 0 > and k = < 0, 0, 1 > .Also, it can be mentioned that if the set of elements of vector space are linearly independent and if every vector in that space can be expressed as linear combination of those elements, then they are said to be as basis for that vector space.Indeed a vector space may have many bases that is any set of linearly independent vectors in some vectors space forms a basis for that space; shortly the base vectors of a vector space are not unique (e.g.linearly independent combinations of them give another base for the same vector space) [12].

V. STANDARD B-SPLINES FOR INTERPOLATION
The main two kinds of local B-Splines concerned with interpolation used in finite strip method are B2-Spline and B3-Spline, where for any one-dimensional field variable () over a partition m of specified interval, the B-Spline interpolant is given by a linear combination of local standard B-splines   () which is associated with each knot of the partition of that interval in addition to extra two knots out of the interval as [13]: ()

B. B3-Spline interpolant series
The equally spaced standard B3-Spline over section length = h, and knot vector [ -2h, -h, 0, h, 2h ], is defined as: The normalized form of this B3-spline is given as: The graph of the B2-Spline, B3-Spline functions as well as a typical series compromised either from them are shown in Fig. 1, Fig. 2 and Fig. 3, respectively:

VI. COMBINED SPLINES FOR INTERPOLATION
In standard B-spline interpolation, the value of field variable at any section knot is a function of the nodal parameters of several adjacent knots this makes a difficulty in treatment of boundary or interior conditions of the plate problem. in order to simplify this drawback, a trails have been focused on retaining the advantage of finite element, that is, restraining a section knot against the field variable will result in a zero value at that knot. the major branch of these trials is to use the combination of properly chosen standard splines.
This kind of interpolant allows the user to introduce interior supports at any section knot as well as easy of handling any type of boundary conditions in much the same way as in a standard finite element formulation, which in turn significantly increases the applicability of spline interpolation in structural analysis.
Since the type of spline series used in standard interpolation is scaled sums of local standard B-splines where they possess a nice properties which some of them are [14], [15]: 1) They possesses translation invariance or periodicity property that is: B k,1 ( + ) = B k,1 (),  ∈ ℝ 2) Their sum also splines 3) Their constant multiple also splines 4) The overall amplitude of it at the center can be forced to any desired value 5) Localized property that is, they have finite values over a number line.
In addition to that, these local spline functions satisfy all the rules listed for vector space, therefore we may look upon those local splines as elements of a vector space, moreover they are as the base vectors of that space.One can use the mentioned discussion to rewrite the standard spline into a more convenient format, which is referred to as a combined spline interpolation.

A. Combined B2-Spline (CB2)
This kind of combination based on the results of references [16], [17], is given in [18], here we review and introduce this combination based on our proposal in order to compare the published integration tables as well as the obtained results in both studies.
For any three equally spaced linearly independent standard B2-Splines associated with successive knots of the partition over section length = h, we have in addition to that central spline derived above, over knot vector: [ -1.5h, -0.5h, 0.5h, 1.5h], the two shifted splines, the first one by 0.5h to the left (i.e. over knot vector [ -2h, -h, 0, h ] ), as : and the second subsequent one which shifted by 0.5h to the right (i.e. over knot vector [ -h, 0, h, 2h ]), as : graph of these successive splines as shown in Fig.  Observing the symmetry about the origin, the linear combination of these three splines can be chosen as: This combined spline can be chosen to have zero values at all knots except its central one, which to be forced to unity, due to symmetry therefore, at knot y = -h: where their normalized forms are: The graph of 2( ) & 2 ′ ( ) for unity of h are shown in Fig. 5 & Fig. 6 :  The last combined splines have 8-internal segments this leads to some computational effort and calculations time, here we propose a new kind of combination which follow the same previous procedures, but will lead to some advantages, as it will be seen later.This kind of combination is made upon standard B3-spline in section 5.B and the two shifted standard B2-splines in section 6.A where the graph of these splines as shown in Fig. 7: with the same requirements which produce the objectives from this operation, therefore the suggested combination take the form: 32() =  ( ) + {( + 0.5ℎ ) + ( − 0.5ℎ ) } Following the same procedure in section 6.1, we obtain the final form of this combination, in addition to its first derivative are shown in the next table : Spline The graph of 32() & 32 ′ () for unity of h are shown in Fig. 8 & Fig. 9 : This new Combined B3B2-Spline functions in addition to retaining all the foregoing discussed nice proposers has only 4-internal segments instead of 8-internal segments for CB2 .
The graph of typical CB2-Spline or CB3B2-Spline series takes the following pattern as shown in Fig. 10:

VII. NUMERICAL EXAMPLES
In order to examine the restudied CB2-Spline interpolation function in connection with comparison to published results based on it as well as to demonstrate the performance of the present combined CB3B2-Spline interpolant in finite strip analysis of plates, the following numerical examples are presented to illustrate the efficiency of the proposed tool, and the main objectives of this study.Moreover, these tools need nothing to do concerned with amendment to satisfy certain boundary condition, since imposing any constraint (boundary condition) can be easily done just by eliminating the row and the column for the pivotal element of the constraint from the overall augmented matrix (i.e.Penalty approach).This nice advantage is due to nodal parameters in conventional spline finite strip becomes really nodal values.

   y
Simply supported square isotropic plate (v = 0.3) with a uniform load, a half plate was analyzed due to symmetry in x-direction (although a quarter of it can also be taken), the boundary conditions and the adapted spline series interpolant for boundary nodes are shown in Fig. 11.

B. Example 2:
Clamped square isotropic plate (v = 0.3) with a uniform load, a half plate was analyzed due to symmetry in xdirection, the boundary conditions and the adapted spline series interpolant for boundary nodes are shown in Fig. 12.

C. Example 3:
Two opposite edges simply supported, the third clamped and the fourth free, rectangular isotropic plate (A × B, B = 2A & v = 0.3),with a uniform load.In this case, a half plate was analyzed because of symmetry in the x-direction.the boundary conditions as well as, the adapted spline series interpolant for boundary nodes are shown in Fig. 13.Knowing that, reference [19], indicate this plate has anticlastic deflection surface near the free edge.The computational results of these examples using (HO3) strips, and the analytical solutions as well as the published results in some literatures based on selected discretization, are given in Table 1, Table 2 and Table 3, where NOS represents the number of strips in x-direction, NOE the number of elementals in y-direction and DOF degrees of freedom.The accuracy as well as the convergence of these values can be observed in addition to comparing with published results Although mathematically, these field variables are not continuous functions of DOF as independent variable, the following plots of these results against DOF gives good visualizations of convergence and comparison, as shown in next figures:       Since the needing of numerical analysis is concerned with the problems that its analytical solutions are not known therefore as the discussed in our paper [21], the best discretization scheme in finite strip method, is to keep the same number of strips as well as the number of harmonics (here is equivalent to elementals) which we call it as systematic discretization.This idea is certified again in the following tables of results for the same foregoing examples in addition to clearing the efficiency of the proposed CB3B2-interpolation function.

VIII. CONCLUSION
The published quadratic combination B2-Spline function (CB2), as well as its implementation in mixed spline finite strip method was reviewed in this article.A farther improvement in this way, that is, the development of combination B3-Spline & B2-Spline functions (CB3B2) was presented.By this method the field variables can be directly worked out with the same order of accuracy and retains all the characteristics of classical finite strip method and Spline finite strip method, but overcomes their drawbacks.It is shown that the advantages of this method are the nodal values were obtained directly without any post calculations, moreover the very simple treatment of boundary conditions and its convenience in analysis is noticeable.Based on this formulation, only the first derivatives of the field variables appear in the functional, so that mere C 0 continuity is required, which guarantees the convergence.The proposed (CB3B2) is still has this feature with some less mathematical requirements for integration performance.Theoretical solutions in the literature were compared with the results of the presented method and the accordance was found to be excellent.It may be noticed that the converges to the accurate solutions are very rapid where usually no high number of degrees of freedom are required as clearly indicated in the tables and figures.Furthermore, the results were superior as compared with some published data which can be seen from the given numerical examples, which indicate, that this proposed work is efficient and accurate.APPENDIX A
these two equations yields:  = 2 &  = −1 2 ⁄ Consequently, this combined B2-Spline (CB2) which works as a base vectors of new vector space, named combined B2-spline vector space, in addition to its first derivative are shown in the next table:

Fig. 10 .
Fig. 10.Typical CB2-Spline or CB3B2-Spline series Since in each elemental the contribution comes only from 4 consecutive local splines, therefore the integration of coupled splines as it is required by the formulation are given in appendixes A, B.

Fig. 11 .
Fig. 11.Simply supported square plate with a uniform load.

TABLE I :
RESULTS OF EXAMPLE 1

TABLE II :
RESULTS OF EXAMPLE 2

TABLE IV :
RESULTS OF EXAMPLE 1 (SYSTEMATIC DISCRETIZATION)

TABLE VI :
RESULTS OF EXAMPLE 3 (SYSTEMATIC DISCRETIZATION)