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Shear flexible curved spline beam element for static analysis
Finite Elements in Analysis and Design 32 (1999) 181}202
Shear #exible curved spline beam element for static analysis M. Ganapathi *, B.P. Patel , J. Saravanan, M. Touratier Institute of Armament Technology, Girinagar, Pune-411025, India Centre for Aeronautics System Studies and Analyses, Bangalore-560 003, India Laboratoire de Modelisation et Mecanique des Structures,URA-1776, CNRS-UPMC Paris 6-ENSAM-ENS Cachan,151, Bd de l'Hopital-75013 Paris, France
Abstract In this paper, an e$cient curved cubic B-spline beam element is developed based on "eld consistency principle, for the static analysis. The formulation is general in the sense that it includes anisotropy, transverse shear deformation, in-plane and rotary inertia e!ects. The element is based on laminated beam theory, which satis"es the interface stress and displacement continuity, and has a vanishing shear stress on the top and bottom surfaces of the beam. The lack of consistency in the shear and membrane strain-"eld interpolations in their constrained physical limits causes poor convergence and unacceptable results due to locking. Hence, numerical experimentation is conducted to check these de"ciencies with a series of assumed shear/membrane strain functions, redistributed in a "eld consistent manner. The performance of the element is assessed by studying the static behavior of a variety of problems ranging from straight beam to circular ring. 1999 Elsevier Science B.V. All rights reserved. Keywords: Spline element; Field consistency; Curved beams; Laminates; Locking; Static
1. Introduction The application of spline functions in the analysis of problems concerning structural mechanics has emerged to be an exciting area of research in the recent past. Due to their piecewise form, smoothness, capacity to handle local phenomena and higher-order continuity, spline functions o!er distinct advantages such as computational e$ciency, #exibility to model di!erent boundary conditions, good accuracy and convergence characteristics, and versatility, etc. Among the spline
* Corresponding author. Fax: #00-91-212-592139. E-mail address: [email protected] (M. Ganapathi) 0168-874X/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 8 7 4 X ( 9 9 ) 0 0 0 1 3 - X
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functions, the functions based on B-spline basis, which can have di!erent order of polynomial, is more in usage for the structural analysis. The study of static and dynamic behavior of various structural elements, using di!erent 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 "nite strip method whereas "nite 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 "nite point method by Zhou and Li [5]. Furthermore, "nite 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 e!ect of shear deformation. However, the application of spline function in conjunction with shear deformation theory has been sparsely treated in the literature [8}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}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 signi"cantly 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 di!erent order of spline functions for the constrained "eld variables [10], by constructing B-spline displacement "eld based on the inspection of the Timoshenko beam mode functions [12], and by introducing modi"ed shear modulii [8,9]. The shear locking behavior in the "nite element analysis of beams, plates and shells has been studied by many authors [13}17] in the context of elements based on conventional polynomial functions for the "eld variables. The more versatile approach, among the available techniques for alleviating locking such as reduced/selective integration scheme and assumed strain "elds, etc., is the "eld-consistent formulation [16,17]. It involves systematically eliminating spurious constraints causing shear/membrane locking in shear #exible "nite 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 "eld-consistent shear#exible 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 "eld consistency to "nd 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.
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2. 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, uI and w at point (x, z) from the median surface are expressed as functions of mid-plane displacement u and w and independent rotation h of the normal in xz plane, as uI(x, z)"u(x) (1#z/R)!zw V(x)#[f (z)#gI(z)] (w V(x)#h(x)), w(x, z)"w(x).
(1)
The functions f (z) and gI(z) are de"ned as f (z)"t/p sin(pz/t)!t/pb cos(pz/t),
(2a)
gI(z)"aIz#bI,
(2b)
where t is the thickness of the laminate. In Eqs. (2a), (2b) coe$cients bI are determined such that displacement component uI is continuous at the interface of the adjacent layers and zero at the mid point of the cross section. Finally, the coe$cients b and aI in Eqs. (2a), (2b) are computed from the requirement that the transverse shear stress pIVX is continuous at the interface of the adjacent layers and vanish at the top and bottom surfaces of the beam. The details of the derivations of constants, b, aI and bI can be found in Refs. [18,19]. The strains in the kth layer, +e,I are written as
e I e I # . 0 e
+e,I"
(3)
The mid-plane strain +e,, bending strain +e, and shear strain +e, in Eq. (3) are written as +e,I"u V#w/R, +e,I"!zw VV#[ f (z)#gI (z)] (w VV#h V)#zu V /R, +e,I"( f X#gI X) (w V#h),
(4a) (4b) (4c)
where the subscript comma denotes the partial derivative with respect to spatial coordinate succeeding it. For a composite laminated beam of layers of thickness tI(k"1, 2, 32), and the ply-angle jI(k"1, 2, 32), the necessary expressions for computing the sti!ness coe$cients, available in the literature [20], are used. The stress}strain relation for kth layer is written as
Q Q I +e,I, Q Q where QGH(i, j"1, 6) are the reduced sti!ness coe$cients of kth layer. The potential energy function ; for a curved beam with length l is given as +p,I"
RI> J (6) [+p,I]2 +e,I b dz dx ! +u w h, + f f m,2 dx, V U I RI where d is the vector of the degrees of freedom associated with the displacement "eld in a "nite element discretization and b is the width of the beam; f , and f are the forces per unit length in V U x and z directions, respectively; m is the moment per unit length of the beam. ;(d)"(1/2)
J
(5)
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3. Element description 3.1. 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 "eld variables u , w and h is the cubic B-spline of equal section length (h), and is given as (Fig. 1) O> O> u " a u, w" b u, G G G G G\ G\
O> h" c u, G G G\
Fig. 1. (a) Typical cubic B-spline. (b) Basis of cubic B-spline expression.
(7)
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in which each local cubic B-spline u has non-zero values over four consecutive sections with the G section-knot x"x as the centre, and is de"ned as G 0, x(x , G\ (x!x ), x )x)x , G\ G\ G\ h#3h(x!x )#3h(x!x )!3(x!x ), x )x)x , 1 G\ G\ G\ G\ G u" (8a) G 6h h#3h(x !x)#3h(x !x)!3(x !x), x )x)x , G> G> G> G G> (x !x), x )x)x , G> G> G> 0, x (x, G> 0, x(x , G\ 3(x!x ), x )x)x , G\ G\ G\ 3h#6h(x!x )!9(x!x ), x )x)x , 1 G\ G\ G\ G u " (8b) G V 6h !3h!6h(x !x)#9(x !x), x )x)x , G> G> G G> !3(x !x), x )x)x , G> G> G> 0, x (x G> a , b , c are the spline parameters. The strain}displacement as well as the stress}strain relationships G G G can be rewritten in terms of spline parameters, after substituting Eq. (7) into Eqs. (4a)}(5). Finally, the variation of the total potential energy given in Eq. (6) leads to the governing equation for the static analysis of the beam as
[[K ]#[K ]#[K ]]+d,"+F,, (9)
where [K ], [K ], [K ] are sti!ness matrices, concerning with membrane, bending and shear,
respectively; and +F, is the consistent load vector. These matrices are evaluated with respect to +d,, where +d, is the generalized vector based on spline parameters only. The displacements u and w, and rotation h at the ends of the spline beam element can be expressed in terms of the spline parameters as 1 1 u " (a #4a #a ), u " (!a #a ); M V 2h \ M 6 \ 1 1 u " (a #4a #a ), u " (!a #a ); O O> MO V 2h O\ O> MO 6 O\ 1 1 w " (b #4b #b ), w " (!b #b ); 6 \ V 2h \ 1 1 w " (b #4b #b ), w " (!b #b ); O 6 O\ O O> O V 2h O\ O> 1 1 h " (c #4c #c ), h " (!c #c ); 6 \ V 2h \ 1 1 h " (c #4c #c ), h " (!c #c ). O 6 O\ O O> O V 2h O\ O>
(10)
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If the element is divided into q equal sections, there will be, all together, [3;(q#3)] spline parameters to de"ne the displacements (u , w) and rotation (h) functions. Instead of the spline parameters a , a , b , b , c and c at the ends of the element and, a , a , b , O O O \ O> \ b , c and c outside the element, one can use the displacements, rotation and their derivaO> \ O> tives at the ends, i.e. u , u , w, w, , h and h, . Thus, within the element, for the present case, there V V V are [3;(q!1)] interior spline parameters which are included in the generalized vector +d,, after the transformation based on (10). In this way, the original generalized vector of spline parameters +d, will now consist of physical coordinates vector (displacements and their derivatives) pertaining to the ends of the element and vector of interior spline parameters. In view of the modi"cation in the original vector of spline parameters, mass and sti!ness matrices shown in (9) are to be updated with respect to the modi"ed generalized vector. This procedure enables us to assemble the element level matrices and introduce the desired boundary conditions, as followed in the standard "nite element procedure. Here, we shall experiment with series of assumed membrane/shear strain functions for a displacement-type cubic B-spline element to derive the optimal membrane/shear strain de"nitions that leave the element free of all problems for most of the boundary conditions. The systematic procedure adopted here, as per the "eld consistency principle, is by establishing the consistency of membrane/shear strain "elds by simply smoothing the w and h to the order of the functions for u and w (Level 1 functions below-the Q-element), by smoothing membrane/shear strain "elds to M V V linear functions (Level 2 functions-the L-element), and by smoothing these strain functions to constant form (Level 3 functions-the C-element). 3.2. O-element } The original element ( xeld-inconsistent element } ISIM) In a conventional approach, the interpolating functions u based on B-spline basis, as de"ned by G Eq. (8a), are cubic and associate four constants within intervals, and are used for de"ning the "eld variables interpolations, see (7). Thus, the membrane and shear strains can be derived from
0,
x(x
, G\ a #a x#a x#a x, x )x)x , G\ G\ a #a x#a x#a x, x )x)x , G\ G u" G a #a x#a x#a x, x )x)x , G G> a #a x#a x#a x, x )x)x , G> G> 0, x (x, G> 0,
(11a)
x(x
, G\ b #b x#b x#b x, x )x)x , G\ G\ b #b x#b x#b x, x )x)x , G\ G w" G b #b x#b x#b x, x )x)x , G G> b #b x#b x#b x, x )x)x , G> G> 0, x (x, G>
(11b)
M. Ganapathi et al. / Finite Elements in Analysis and Design 32 (1999) 181}202
x(x
, G\ c #c x#c x#c x, x )x)x , G\ G\ c #c x#c x#c x, x )x)x , G\ G h" G c #c x#c x#c x, x )x)x , G G> c #c x#c x#c x, x )x)x , G> G> 0, x (x. G> as
187
0,
(11c)
x(x
0,
, G\ (a #b /R)#(2a #b /R)x#(3a #b /R)x#(b /R)x, x )x)x , G\ G\ (a #b /R)#(2a #b /R)x#(3a #b /R)x#(b /R)x, x )x)x , G G> eN" G (a #b /R)#(2a #b /R)x#(3a #b /R)x#(b /R)x, x )x)x , G G> (a #b /R)#(2a #b /R)x#(3a #b /R)x#(b /R)x, x )x)x , G> G> 0, x (x, G> (12a)
x(x , G\ (b #c )#(2b #c )x#(3b #c )x#(c )x, x )x)x , G\ G\ (b #c )#(2b #c )x#(3b #c )x#(c )x, x )x)x , G\ G eQ"( f, #g,I) G X X (b #c )#(2b #c )x#(3b #c )x#(c )x, x )x)x , G G> (b #c )#(2b #c )x#(3b #c )x#(c )x, x )x)x , G> G> 0, x (x. G> (12b) 0,
If the element is to model a situation in which in-extensional bending and the Kirchho! limit of classical thin beam predominate, the requirements that the membrane and shear strain should vanish, produce in e!ect the following conditions: Membrane case: (a #b /R)"0, (a #b /R)"0, (a #b /R)"0, (a #b /R)"0,
(2a #b /R)"0, (2a #b /R)"0, (2a #b /R)"0, (2a #b /R)"0,
(3a #b /R)"0, (3a #b /R)"0, (3a #b /R)"0, (3a #b /R)"0,
(b /R)"0, (b /R)"0, (b /R)"0, (b /R)"0,
x )x)x , G\ G\ x )x)x , G\ G x )x)x , G G> x )x)x . G> G> (13a)
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Shear case: (b #c )"0, (b #c )"0, (b #c )"0, (b #c )"0,
(2b #c )"0, (2b #c )"0, (2b #c )"0, (2b #c )"0,
(3b #c )"0, (3b #c )"0, (3b #c )"0, (3b #c )"0,
(c )"0, (c )"0, (c )"0, (c )"0,
x
)x)x , G\ G\ x )x)x , G\ G x )x)x , G G> x )x)x. G>
(13b)
Using the element, based on the original interpolation functions employing directly in deriving the constrained strain "elds, our studies here show that this element locks (i.e. spurious over-sti!ening that increases inde"nitely with reduction in beam thickness). As per "eld consistency paradigm, the derivative functions for u and w do not match that for the w and h in the membrane and shear V V strain de"nitions in precisely these cubic terms. Field consistency requires, in a simple sense, that the membrane/shear strain must be interpolated in a consistent manner. Thus, w term in e, and h in e must be consistent with the "eld functions for u and w , respectively. M V V 3.3. Q-element } the element based on Level 1 consistency (CMCS-1) At the simplest level of "eld-consistency, we consider the use of "eld redistributed substitute interpolation functions which include only those speci"c terms that must be made "eld-consistent, as outlined in Refs. [16,17]. This is achieved here by smoothing the original interpolation function in a least-squares-accurate fashion to the desired form, i.e. the functions that are consistent with the derivative functions. Here, we need smoothed functions for w and h which are consistent with the interpolations u , and w for the use in the membrane and shear strains de"nitions, respectively. It M V V can be noted that u and w are of the quadratic form, as seen from (8b) in de"ning the function M V V u which is involved in evaluating the derivative of the "eld variables, as per O-level description. G V This means that the smoothed functions we must derive for w, and h are, designated as u, obtained G by smoothing the original functions u to be a least-squares form consistent with the derivative G functions u . This operation is simple and the substitute interpolation functions obtained, in this G V way, are given as follows:
0,
x(x
, G\ 1/120!1/10(x!x )/h#(1/4) (x!x )/h, x )x)x , G\ G\ G\ G\ 17/120#(4/5) (x!x )/h!(x!x )/4h, x )x)x , G\ G\ G\ G u" G 83/120!(36/120) (h!x #x)/h!(30/120) (h!x #x)/h, x )x)x , G> G> G G> 19/120!(2/5) (h!x #x)/h#(1/4) (h!x #x)/h, x )x)x , G> G> G> G> 0, x (x. G> (14) The functions u is retained for u and w in the membrane and shear strains computation. G V M V V However, we shall see below that this element can still lock for certain sets of boundary conditions, even though consistent de"nitions for the membrane and shear strain "elds have been assured within the element domain. This suggests that an inconsistency in the shear/membrane strain
M. Ganapathi et al. / Finite Elements in Analysis and Design 32 (1999) 181}202
189
de"nitions have been introduced for some set of boundary conditions. It is therefore necessary to examine lower levels of consistency based on further sets of assumed membrane/shear strain functions to see if these locking mechanisms/de"ciency can be removed.
3.4. L-element } the element based on Level 2 consistency (CMCS-2) It is logical now to consider smoothed interpolation functions based on linear form for w and h pertaining to membrane and shear strains, respectively. This will necessitate that the derivative functions de"ning u and w will also have to be smoothed to linear form. The shape functions M V V required here can be denoted u for w and h, and u for u and w , respectively. The G G V M V V interpolation functions u are obtained by smoothing the original derivative forms u in G V G V a least-squares-accurate operation over the element domain. These functions are then
x(x , G\ !1/30#(3/20) (x!x )/h, x )x)x , G\ G\ G\ 11/60#(11/20) (x!x )/h, x )x)x , G\ G\ G u" G 11/15!(11/20) (h!x #x)/h, x )x)x , G> G G> 7/60!(3/20) (h!x #x)/h, x )x)x , G> G> G> 0, x (x. G>
(15a)
x(x , G\ !1/(12h)#(1/2h) (x!x )/h, x )x)x , G\ G\ G\ 3/4h!(1/2h) (x!x )/h, x )x)x , G\ G\ G u " G V !1/4h!(1/2h) (h!x #x)/h, x )x)x , G> G G> !5/12h!(1/2h) (h!x #x)/h, x )x)x , G> G> G> 0, x (x. G>
(15b)
0,
0,
Numerical experiments below show that this element is identical in behavior as seen in the case of Level 1, and it locks for most of the practical boundary conditions for a beam. Thus smoothing to level 2 has failed to remove the locking mechanism.
3.5. C-element } the element based on Level 3 consistency (CMCS-3) We now go down to the lowest level (a constant form for representing the w and h) and correspondingly, the derivative functions de"ning u , and w will also have to be smoothed to M V V constant form. The interpolation functions for this case is assigned as u for w and h, and u for G G V u and w , respectively. The functions u are obtained by smoothing the original derivative M V V G V forms u in a least-squares accurate operation over the element domain. It can be shown that G V
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these functions to be used are
x(x , G\ 1/24, x )x)x , G\ G\ 11/24, x )x)x , G\ G u" G 11/24, x )x)x , G G> 1/24, x )x)x , G> G> 0, x (x, G> 0,
0,
(16a)
x(x
, G\ 1/6h, x )x)x , G\ G\ 1/2h, x )x)x , G\ G u " G V !1/2h, x )x)x , G G> !1/6h, x )x)x , G> G> 0, x (x. G>
(16b)
The numerical analysis below shows that this element does not lock for the boundary conditions considered here, i.e. its accuracy is insensitive to a very large variation in the thickness parameter of the beam. It can be noted that this can be achieved by using a reduced integration rule, i.e. a Gaussian one-point integration of membrane and shear energy will produce the same results. However, for the thick beam, the numerical study given below brings out di!erent results compared to the actual one. The procedure adopted by "eld consistency approach permits greater #exibility in the choice of integration order for the energy terms, as these can now be determined by the other considerations (such variable thickness, more accurate integration of 1/R terms, etc.) and not dictated by the need for "eld consistency.
4. 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 di!erent functions for the redistributed strain "elds through the "eld consistent strategy. 4.1. Test case 1: clamped}clamped isotropic circular arc with sector angle "1803 The numerical results for the static analyses are evaluated using the element developed in the earlier section and are shown in Table 1. The results obtained here are the maximum de#ection of the arc having slenderness ratio of ¸/t"10, 100 and 10. They are also in the form of convergence studies for de#ections (w "w [EI/PR], where P is the load at the centre of the arc, and EI is the
M. Ganapathi et al. / Finite Elements in Analysis and Design 32 (1999) 181}202
191
Table 1 Non-dimensional de#ection at the centre of clamped}clamped circular arc with sector angle 1803 with di!erent consistent strategies (M: Membrane, S: Shear, O: Original, Q: Quadratic, L: Linear, C: Constant, RI: Reduced Integration, CL: Classical Theory) q ¸/t 10
100
10
MO
MQ
ML
MC
MO
MQ
ML
MC
MO
MQ
ML
MC
4 SO SQ SL SC RI
0.01916 0.01923 0.02003 0.02050 0.11571
0.01921 0.01929 0.02009 0.02057
0.01993 0.02001 0.02089 0.02142
0.02106 0.02116 0.02229 0.02296
0.00033 0.00034 0.00037 0.00058 0.01264
0.00033 0.00034 0.00037 0.00059
0.00036 0.00037 0.00040 0.00068
0.00050 0.00054 0.00064 0.00806
0.00000 0.00000 0.00000 0.00000 0.01123
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00790
6 SO SQ SL SC RI
0.02506 0.02508 0.02549 0.02584 0.10069
0.02508 0.02509 0.02550 0.02585
0.02517 0.02518 0.02559 0.02595
0.02556 0.02558 0.02601 0.02635
0.00393 0.00411 0.00435 0.00615 0.01163
0.00402 0.00421 0.00446 0.00637
0.00426 0.00448 0.00476 0.00694
0.00538 0.00575 0.00623 0.01085
0.00000 0.00000 0.00000 0.00000 0.01048
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.01068
8 SO SQ SL SC RI
0.02595 0.02595 0.02622 0.02679 0.10479
0.02595 0.02595 0.02622 0.02679
0.02602 0.02602 0.02629 0.02686
0.02624 0.02625 0.02652 0.02709
0.00913 0.00926 0.00945 0.01046 0.01244
0.00921 0.00934 0.00954 0.01055
0.00932 0.00945 0.00965 0.01070
0.00974 0.00989 0.01011 0.01143
0.00000 0.00000 0.00000 0.00000 0.01127
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.01126
10 SO SQ SL SC RI
0.02628 0.02628 0.02644 0.02704 0.10014
0.02628 0.02629 0.02645 0.02704
0.02632 0.02632 0.02648 0.02708
0.02652 0.02652 0.02668 0.02728
0.01089 0.01093 0.01101 0.01139 0.01250
0.01091 0.01096 0.01104 0.01142
0.01094 0.01099 0.01107 0.01146
0.01105 0.01110 0.01118 0.01163
0.00000 0.00000 0.00000 0.00000 0.01140
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.01147
14 SO SQ SL SC RI
0.02660 0.02660 0.02669 0.02718 0.09752
0.02660 0.02660 0.02669 0.02718
0.02662 0.02662 0.02671 0.02721
0.02675 0.02675 0.02685 0.02734
0.01162 0.01163 0.01166 0.01174 0.01263
0.01162 0.01163 0.01166 0.01174
0.01163 0.01164 0.01166 0.01174
0.01164 0.01165 0.01167 0.01176
0.00000 0.00000 0.00000 0.00000 0.01157
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.01159
16 SO SQ SL SC RI
0.02668 0.02668 0.02675 0.02720 0.09669
0.02668 0.02668 0.02675 0.02720
0.02670 0.02670 0.02677 0.02722
0.02682 0.02682 0.02689 0.02733
0.01171 0.01172 0.01173 0.01178 0.01265
0.01171 0.01172 0.01173 0.01178
0.01171 0.01172 0.01173 0.01178
0.01172 0.01172 0.01174 0.01178
0.00000 0.00000 0.00000 0.00000 0.01160
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.01162
20 SO SQ SL SC RI
0.02679 0.02679 0.02684 0.02720 0.09423
0.02679 0.02679 0.02684 0.02720
0.02680 0.02680 0.02685 0.02721
0.02690 0.02690 0.02694 0.02731
0.01178 0.01178 0.01179 0.01180 0.01265
0.01178 0.01178 0.01179 0.01180
0.01178 0.01178 0.01179 0.01180
0.01178 0.01179 0.01179 0.01181
0.00000 0.00000 0.00000 0.00000 0.01163
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.01164
24 SO SQ SL SC RI
0.02686 0.02686 0.02689 0.02719 0.09240
0.02686 0.02686 0.02689 0.02719
0.02687 0.02687 0.02690 0.02720
0.02695 0.02695 0.02698 0.02728
0.01180 0.01181 0.01181 0.01181 0.01264
0.01180 0.01181 0.01181 0.01181
0.01181 0.01181 0.01181 0.01182
0.01181 0.01181 0.01181 0.01182
0.00000 0.00000 0.00000 0.00000 0.01164
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.01165
CL 0.01166
0.01166
0.01166
192
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#exural rigidity) with the number of B-spline sections q. The solutions obtained based on the original functions for the "eld variables (u, w, h), which results in "eld inconsistency approach for the constrained strains (membrane and shear), and also employing di!erent levels "eld consistent representation for the constrained strain functions (quadratic, linear, constant) are indicated in the table. The comparative results quoted in the table are calculated using one point numerical reduced integration scheme (RI) for the evaluation of membrane and shear energy terms considering the original functions, and the analytical solutions within the con"nes of Euler}Bernoulli beam theory (CL). It can be noted here that the numerical experiments with various combinations of reduced integration techniques for estimating the membrane and shear energy terms were conducted. The best accuracy was attained for one-point quadrature scheme for both membrane and shear energy terms. For the sake of brevity, the detailed study is not presented here. It is very clear from Table 1 that the rate of convergence is very good for both thick and thin beams with the increase in the spline sections. It may be inferred from Table 1 that for fairly thick and thin cases (¸/t"10, 100), the results obtained adopting "eld inconsistency manner are in close agreement with those corresponding to the di!erent levels of "eld consistent strategy for the constrained strain functions. However, the reduced integration scheme yields erroneous results for fairly thick beam ¸/t"10, but it gives acceptable results for fairly as well as extremely thin situations as highlighted in Table 1. Furthermore, it is amply clear from the table that, with the reasonable number of spline sections q for modeling the very thin beam (¸/t"10) adequately, the element exhibits severe locking when the full integration scheme is employed. With lowering the order of level for the "eld consistency requirement in de"ning the constrained strain terms from quadratic to linear function does not alleviate the locking. It may be inferred from the table that the consistent way of representing the constrained strains (membrane and shear) at constant level produces remarkable recovery from the locking syndrome. However, applying the consistent strategy either for shear or membrane alone cannot free the spurious energy locked in the element. The numerical experiment conducted here, in general, suggests that for the beam thickness parameter ¸/t of the order 100, the original spline element does not seem to su!er much from locking problems whereas it is susceptible to membrane and shear locking for the extreme thin case. Furthermore, this study again reveals that unlike element employing the reduced integration scheme, which may not be suitable for the problems wherein a higher order of integration is needed (for instance tapered beams), consistent element frees the order of integration and thus, is applicable for both thick and extreme thin situations, as expected [16,17] The results obtained using the consistent formulation for the extremely thin beam agree very well with the classical solutions. The stress distributions obtained with inconsistency approach, and the consistency formulation with constant functions for the constrained strains are given in Fig. 2 (q"24, ¸/t"10). For the inconsistency approach (IMIS), one can notice the completely erroneous results for axial thrust, bending moment and shear stress, i.e. large values for axial and shear stresses, and very small values for moment are the typical of a locking solution. But, the stresses evaluated based on consistent formulation (CMCS-3) yield accurate distributions and match well with the analytical results. 4.2. Test case 2: clamped-free isotropic circular arc with sector angle "1803 A clamped-free circular arc loaded at the free end is considered here. De#ections obtained at the free end (w "w [EI/PR]) are given in Table 2 by varying the aspect ratio of the beam from fairly D
M. Ganapathi et al. / Finite Elements in Analysis and Design 32 (1999) 181}202
193
Fig. 2. Stress and moment distributions for clamped}clamped isotropic are using consistent and inconsistent models: (a) axial stress, (b) bending moment, (c) shear stress.
thick one to extreme thin situation. The di!erent results calculated considering full and reduced integration schemes are highlighted along with those of the "eld consistent formulation at di!erent levels of strategies for the membrane and shear strain "elds. Like test case 1, one can infer from Table 2 (extreme thin) that the element exhibits signi"cant locking, while performing the full
194
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Table 2 Non-dimensional de#ection at free end of clamped-free circular arc with sector angle 1803 with di!erent consistent strategies (M: Membrane, S: Shear, O: Original, Q: Quadratic, L: Linear, C: Constant, RI: Reduced Integration, CL: Classical Theroy) q ¸/t 10
100
10
MO
MQ
ML
MC
MO
MQ
ML
MC
MO
MQ
ML
MC
2 SO SQ SL SC RI
1.41439 1.41801 1.43074 1.76082 11.7550
1.43920 1.44309 1.45318 1.76278
1.54838 1.55680 1.56201 1.79268
2.27168 2.44178 4.23596 23.5202
0.89325 0.89397 0.90173 1.68494 11.5980
0.92233 0.92375 0.93361 1.68495
1.12957 1.21052 1.47907 1.68853
1.32338 1.34737 3.18143 23.4813
0.84634 0.84657 0.84647 1.68492 11.4935
0.84641 0.84642 0.84628 1.68536
0.84609 0.84607 1.47727 1.68749
1.29180 1.29228 3.17112 23.7800
4 SO SQ SL SC RI
1.61428 1.61447 1.61587 1.61787 2.14890
1.61550 1.61569 1.61712 1.61897
1.61644 1.61663 1.61810 1.61985
1.76314 1.77625 1.94809 3.65888
1.34129 1.34562 1.34830 1.37467 2.00516
1.38385 1.38977 1.39337 1.40769
1.42978 1.44312 1.46463 1.52218
1.55748 1.57900 1.64886 3.60176
0.84629 0.84634 0.84630 1.12992 2.00334
0.84648 0.84648 0.84622 1.12973
0.84626 0.84614 0.85679 1.50110
1.28457 1.28543 1.63586 3.62173
6 SO SQ SL SC RI
1.62194 1.62196 1.62243 1.62420 1.82209
1.62204 1.62207 1.62254 1.62431
1.62209 1.62211 1.62258 1.62436
1.68624 1.69163 1.76199 2.17560
1.53418 1.53540 1.53582 1.53963 1.67697
1.54221 1.54348 1.54388 1.54817
1.54475 1.54606 1.54642 1.55451
1.57657 1.57859 1.58588 2.11737
0.84699 0.84646 0.84691 1.01513 1.67582
0.84662 0.84649 0.84652 1.01507
0.84592 0.84599 1.30807 1.43537
1.28082 1.28044 1.46295 2.11221
8 SO SQ SL SC RI
1.62278 1.62279 1.62307 1.62449 1.75945
1.62280 1.62281 1.62309 1.62451
1.62281 1.62282 1.62309 1.62452
1.66182 1.66492 1.70200 1.85457
1.56458 1.56486 1.56498 1.56567 1.61355
1.56613 1.56641 1.56652 1.56728
1.56639 1.56677 1.56678 1.56780
1.57498 1.57546 1.57909 1.79701
0.84590 0.84570 0.84581 0.96485 1.61223
0.84584 0.84577 0.84623 0.96494
0.84530 0.84533 1.22990 1.34657
1.27988 1.27998 1.39740 1.79728
10 SO SQ SL SC RI
1.62309 1.62310 1.62329 1.62444 1.73921
1.62310 1.62310 1.62330 1.62445
1.62310 1.62310 1.62330 1.62445
1.64989 1.65185 1.67328 1.74272
1.56960 1.56968 1.56971 1.56987 1.59285
1.57000 1.57008 1.57012 1.57029
1.57005 1.57013 1.57016 1.57036
1.57360 1.57381 1.57612 1.68614
0.84625 0.84666 0.84626 0.93882 1.59135
0.84647 0.84677 0.84647 0.93853
0.84559 0.84716 1.16963 1.27676
1.27844 1.27752 1.36217 1.68607
14 SO SQ SL SC RI
1.62339 1.62339 1.62351 1.62431 1.72651
1.62339 1.62339 1.62351 1.62431
1.62339 1.62339 1.62351 1.62431
1.63826 1.63916 1.64743 1.66817
1.57110 1.57111 1.57112 1.57114 1.57966
1.57116 1.57117 1.57117 1.57119
1.57116 1.57117 1.57118 1.57120
1.57239 1.57247 1.57365 1.61298
0.84691 0.84638 0.84601 0.91001 1.57622
0.84652 0.84634 0.84670 0.90996
0.84567 0.84585 1.08779 1.17342
1.28851 1.28718 1.34086 1.62034
16 SO SQ SL SC RI
1.62348 1.62348 1.62357 1.62426 1.72422
1.62348 1.62348 1.62357 1.62426
1.62348 1.62348 1.62357 1.62426
1.63507 1.63571 1.64111 1.65395
1.57123 1.57123 1.57124 1.57125 1.57721
1.57125 1.57126 1.57126 1.57127
1.57125 1.57126 1.57126 1.59127
1.57211 1.57218 1.57308 1.59920
0.84263 0.84267 0.84277 0.89857 1.57459
0.84278 0.84296 0.84414 0.89816
0.84480 0.84302 1.06083 1.13468
1.26197 1.26467 1.31036 1.59353
20 SO SQ SL SC RI
1.62359 1.62359 1.62366 1.62419 1.72201
1.62359 1.62359 1.62366 1.62419
1.62359 1.62359 1.62366 1.62419
1.63105 1.63139 1.63389 1.63973
1.57130 1.57130 1.57131 1.57131 1.57481
1.57131 1.57131 1.57131 1.57132
1.57131 1.57131 1.57131 1.57132
1.57181 1.57185 1.57243 1.58559
0.84680 0.84639 0.84607 0.88993 1.57408
0.84669 0.84814 0.84662 0.88856
0.84800 0.84825 1.02462 1.08816
1.27178 1.28896 1.30153 1.57513
CL 1.570796
1.570796
1.570796
M. Ganapathi et al. / Finite Elements in Analysis and Design 32 (1999) 181}202
195
Fig. 3. Stress and moment distributions for clamped-free isotropic arc using consistent and inconsistent models: (a) axial stress, (b) bending moment, (c) shear stress.
integration scheme for evaluating the sti!ness matrices. By lowering the order of level for the "eld consistency requirement in de"ning the constrained strain terms from quadratic to linear function, locking to some extent is getting freed but the spurious energy is completely removed for the constrained strains with constant functions. The qualitative trend of the results obtained using "eld
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M. Ganapathi et al. / Finite Elements in Analysis and Design 32 (1999) 181}202
inconsistency as well as consistency manner is in general, similar to that of Test case 1. The results based on "eld consistency manner once again amply demonstrate that it is in very good agreement with those of the original functions adopting full integration scheme for the thick situation and they match well with the classical solution for the thin cases. The stress oscillations due to inconsistency formulation, and the actual stress distributions obtained using the consistency formulation with constant functions for the constrained strains are given in Fig. 3 (q"24, ¸/t"10). For the inconsistency approach (IMIS), one can notice the stress oscillation, around the free end at which load is applied, and erroneous results (also large values for axial and shear stresses are the typical of a locking solution). But, the stresses evaluated using the consistent formulation (CMCS-3) match well with the analytical results. 4.3. Test case 3: pinched circular ring The static analysis of isotropic pinched ring is carried out using half-ring model idealization. The variation of the de#ection at the loading point w "2w [EI/PR] with the aspect ratio, spline section and di!erent consistent strategies is described in Table 3. The element behavior for this problem, in general, is quite similar to those of Test case 1. However, unlike the previous test cases, it is apparently visible from Table 3 that the results evaluated for thin ring adopting reduced integration scheme is di!erent from the classical solutions and it seems to decrease with the increase in the spline sections. But, the "eld-consistent element based on constant level approximation alone for the spurious energy terms behaves very well for very thin beam and the results are in very good agreement with Euler}Bernoulli analysis. Fig. 4 (q"24, ¸/t"10) shows the stress oscillations, near the symmetric boundary conditions (also large values for axial and shear stresses, and very small values for moment are the typical of a locking solution), due to the inconsistency formulation. The stress distributions evaluated using the consistency formulation (CSCM-3) match very well with the existing solutions. Like, the previous examples, the consistent formulation ensures the accurate results. 4.4. Test case 4: analysis of simply supported composite laminates with sinusoidal load [P sin(px/l)] To see e$cacy of the element formulation, the problems of straight cross-ply (03, 03/903, 03/903/03) laminated beams, and symmetric cross-ply curved laminates (03/903/03) are considered using number of spline sections q"24. The material properties used here are E /E "25, G /E "0.5, G /E "0.2, G /E "0.5, l "0.25. The value of the displacement for the straight beam case is plotted against the aspect ratio in Fig. 5. The formulation based on CSCM-3 alone yields accurate results and compare very well with exact solution. The variation of stresses through the thickness of the laminate is shown in Fig. 6 along with the exact solution. They are found to be in good agreement. Similar study is made for the curved cross-ply composite laminates (03/903/03), having sector angle 603 and subjected to sinusoidal load, for di!erent R/t. Table 4 gives displacement and stresses obtained from the present formulation and compares with elasticity solution.
M. Ganapathi et al. / Finite Elements in Analysis and Design 32 (1999) 181}202
197
Table 3 Non-dimensional de#ection (= "2w EI/PR) at the loading point of pinched ring with di!erent consistent strategies (M: M Q: Quadratic, L: Linear, C: Constant, RI: Reduced Integration, CL: Classical Theroy) Membrane, S: Shear, O: Original, q ¸/t 10
100
10
MO
MQ
ML
MC
MO
MQ
ML
MC
MO
MQ
ML
MC
2 SO SQ SL SC RI
0.15034 0.15047 0.15087 0.15112 434.604
0.15187 0.15201 0.15242 0.15268
0.15370 0.15387 0.15430 0.15465
0.23684 0.25872 0.25951 3.36113
0.05242 0.05245 0.05245 0.05257 434.485
0.08130 0.08133 0.08133 0.08152
0.14274 0.14293 0.14293 0.14789
0.17402 0.17426 0.17427 3.35486
0.00000 0.00000 0.00000 0.00000 461.004
0.00000 0.00000 0.00000 0.00000
0.14217 0.14216 0.14214 0.14781
0.17324 0.17325 0.17329 2.96077
4 SO SQ SL SC RI
0.15508 0.15509 0.15519 0.15538 121.319
0.15511 0.15512 0.15521 0.15541
0.15512 0.15512 0.15522 0.15542
0.16516 0.16566 0.16652 0.42411
0.14146 0.14162 0.14163 0.14209 121.025
0.14362 0.14379 0.14380 0.14427
0.14411 0.14430 0.14430 0.14493
0.15193 0.15225 0.15227 0.41751
0.00000 0.00000 0.00000 0.00000 115.192
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.12479
0.14549 0.14550 0.14550 0.39485
6 SO SQ SL SC RI
0.15524 0.15524 0.15529 0.15547 76.9276
0.15524 0.15524 0.15529 0.15547
0.15524 0.15524 0.15529 0.15547
0.15958 0.15977 0.16007 0.22198
0.14809 0.14812 0.14813 0.14821 76.4563
0.14829 0.14832 0.14832 0.14840
0.14831 0.14834 0.14834 0.14843
0.14979 0.14984 0.14986 0.21514
0.00000 0.00000 0.00000 0.00000 72.0788
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.09689
0.14312 0.14311 0.14310 0.21663
8 SO SQ SL SC RI
0.15529 0.15529 0.15532 0.15547 57.7407
0.15529 0.15529 0.15532 0.15547
0.15529 0.15529 0.15533 0.15547
0.15788 0.15800 0.15817 0.18152
0.14871 0.14872 0.14872 0.14873 57.1306
0.14874 0.14875 0.14875 0.14877
0.14875 0.14875 0.14875 0.14877
0.14929 0.14931 0.14932 0.17457
0.00000 0.00000 0.00000 0.00000 60.1938
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.07653
0.14252 0.14256 0.14260 0.17297
10 SO SQ SL SC RI
0.15532 0.15532 0.15535 0.15546 46.4402
0.15532 0.15532 0.15535 0.15546
0.15532 0.15532 0.15535 0.15547
0.15709 0.15716 0.15727 0.16868
0.14881 0.14881 0.14881 0.14882 45.7038
0.14882 0.14882 0.14882 0.14882
0.14882 0.14882 0.14882 0.14882
0.14909 0.14910 0.14911 0.16175
0.00001 0.00001 0.00001 0.00001 54.5013
0.00001 0.00001 0.00001 0.00001
0.00001 0.00001 0.00001 0.06260
0.14237 0.14235 0.14226 0.16165
14 SO SQ SL SC RI
0.15536 0.15536 0.15537 0.15546 33.6433
0.15536 0.15536 0.15537 0.15546
0.15536 0.15536 0.15537 0.15546
0.15635 0.15638 0.15643 0.16034
0.14884 0.14884 0.14884 0.14884 32.6745
0.14884 0.14884 0.14884 0.14884
0.14884 0.14884 0.14884 0.14884
0.14895 0.14895 0.14896 0.15352
0.00006 0.00006 0.00006 0.00006 30.4532
0.00008 0.00008 0.00008 0.00009
0.00008 0.00008 0.00008 0.04558
0.14278 0.14283 0.14295 0.15420
16 SO SQ SL SC RI
0.15537 0.15537 0.15538 0.15545 29.6797
0.15537 0.15537 0.15538 0.15545
0.15537 0.15537 0.15538 0.15545
0.15615 0.15617 0.15621 0.15875
0.14884 0.14884 0.14884 0.14884 28.6074
0.14884 0.14884 0.14884 0.14884
0.14884 0.14884 0.14884 0.14884
0.14892 0.14892 0.14893 0.15198
0.00013 0.00013 0.00013 0.00014 23.2414
0.00018 0.00018 0.00018 0.00020
0.00018 0.00018 0.00018 0.04013
0.14208 0.14201 0.14223 0.15185
20 SO SQ SL SC RI
0.15538 0.15538 0.15539 0.15545 24.16929
0.15538 0.15538 0.15539 0.15545
0.15538 0.15538 0.15539 0.15545
0.15589 0.15591 0.15593 0.15716
0.14884 0.14884 0.14884 0.14884 22.91724
0.14884 0.14884 0.14884 0.14884
0.14884 0.14884 0.14884 0.14884
0.14889 0.14889 0.14889 0.15045
0.00047 0.00048 0.00048 0.00053 28.17586
0.00068 0.00068 0.00069 0.00075
0.00069 0.00069 0.00069 0.03268
0.14227 0.14214 0.14171 0.14992
24 SO SQ SL SC RI
0.15539 0.15539 0.15539 0.15544 20.52279
0.15539 0.15539 0.15539 0.15544
0.15539 0.15539 0.15539 0.15544
0.15575 0.15576 0.15577 0.15645
0.14884 0.14884 0.14884 0.14885 19.12632
0.14884 0.14884 0.14884 0.14885
0.14884 0.14884 0.14884 0.14885
0.14887 0.14887 0.14888 0.14977
0.0014 0.00142 0.00142 0.00157 18.01243
0.002 0.00201 0.00201 0.00222
0.00201 0.00203 0.00203 0.02849
0.14234 0.14227 0.14171 0.14942
CL 0.148778
0.148778
0.148778
198
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Fig. 4. Stress and moment distributions for isotropic pinched circular ring using consistent and inconsistent models: (a) axial stress, (b) bending moment, (c) shear stress.
Fig. 7 presents a comparison for the variation of shear stress distribution along the thickness with those of elasticity solutions. The results are, in general, very good agreement with the analytical solution.
M. Ganapathi et al. / Finite Elements in Analysis and Design 32 (1999) 181}202
199
Fig. 5. Normalized transverse displacement with aspect ratio for cross-ply straight beam.
Fig. 6. Stress distribution in straight cross-ply beam: (a) normalized shear stress, (b) normalized in-plane stress.
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Fig. 7. Normalized transverse shear stress distribution in curved cross-ply (R/t"20, 50).
M. Ganapathi et al. / Finite Elements in Analysis and Design 32 (1999) 181}202
201
Table 4 Comparison of results with elasticity solution (ES) for cross-ply simply supported laminated arc ( "603) R/t
20 50 100 1000 10000
pH V
wH Present
ES
Present
ES
2.3767 2.0245 1.9690 1.9452 1.9442
2.38 2.02 1.96 } }
!8.3986 !7.9440 !7.8467 !7.7820 !7.7767
!8.5 !7.98 !7.87 } }
Note: wH"10 E w(¸/2, 0)t/p R pH"10 p (¸/2, !t/2)t/p R. * M V V M
5. Conclusions The shear-membrane locking phenomena in the shear #exible curved element based on cubic B-spline functions for the displacement "elds have been analyzed using the "eld-consistency approach. The "eld-consistent redistribution of membrane and shear strain "elds 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 e!ectiveness of the element has been demonstrated here for the static analysis of a wide range of problems without membrane or shear locking e!ects emerging.
References [1] Y.K. Cheung, S.C. Fan, C.Q. Wu, Spline "nite strip in structural analysis, Proceedings of the International Conference. On Finite Element Method, Shanghai, 1982, pp. 704}709. [2] Y.K. Cheung, S.C. Fan, Static analysis of right box girder bridges by "nite strip method, Proc. Inst. Civ. Eng. Part 2 75 (1983) 311}323. [3] C.T. Shik, On spline "nite element, J. Comput. Math. 1 (1979) 50}72. [4] A. Gupta, J. Kiusalaas, M. Saraph, Cubic B-spline for "nite element analysis of axisymmetric shells, Comput. Struct. 38 (1991) 463}468. [5] H.B. Zhou, G.Y. Li, Free vibration analysis of sandwich plates with laminated faces using spline "nite point method, Comput. Struct. 59 (1996) 257}263. [6] A.Y.T. Leung, F.T.K. Au, Spline "nite elements for beam and plate, Comput. Struct. 37 (1990) 717}729. [7] S.C. Fan, M.H. Luah, Free vibration analysis of arbitrary thin shell structures by using spline "nite element, J. Sound Vib. 179 (1995) 763}776. [8] I. Patlashenko, T. Weller, Two-dimensional spline collocation method for nonlinear analysis of laminated panels, Comput. Struct. 57 (1995) 131}139. [9] I. Patlashenko, T. Weller, Cubic B-spline collocation method for nonlinear static analysis of panels under mechanical and thermal loadings, Comput. Struct. 49 (1993) 89}96. [10] D.J. Dawe, S. Wang, Vibration of shear-deformable beams using a spline-function approach, Int. J. Numer. Methods Eng. 33 (1992) 819}844. [11] S. Wang, D.J. Dawe, Vibration of shear-deformable rectangular plates using a spline-function Rayleigh-Ritz approach, Int. J. Numer. Methods Eng. 36 (1993) 695}711.
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[12] S. Wang, A uni"ed Timoshenko beam B-spline Rayleigh-Ritz method for vibration and buckling analysis of thick and thin beams and plates, Int. J. Numer. Methods Eng. 40 (1997) 473}491. [13] E.D.L. Pugh, E. Hinton, O.C. Zienkiewicz, A study of quadrilateral plate bending elements with reduced integration, Int. J. Numer. Methods Eng. 12 (1978) 1059}1079. [14] D.S. Malkus, T.J.R. Hughes, Mixed "nite element methods-reduced and selective integration techniques: a uni"cation of concepts, Comput. Methods Appl. Mech. Eng. 15 (1978) 63}81. [15] M. Ganapathi, O. Polit, M. Touratier, A C3 eight-node membrane-shear-bending element for geometrically non-linear (static and dynamic) analysis of laminates, Int. J. Numer. Methods Eng. 39 (1996) 3453}3474. [16] G. Prathap, C. Ramesh Babu, Field-consistency and violent stress oscillations in the "nite method, Int. J. Numer. Methods Eng. 24 (1987) 2017}2033. [17] G. Prathap, B.P. Naganarayana, B.R. Somashekar, Field-consistency analysis of the isoparametric eight-noded plate bending element, Comput. Struct. 29 (1988) 857}873. [18] A. Beakou, M. Touratier, A rectangular "nite element for analysing composite multilayered shallow shells in statics, vibration and buckling, Int. J. Numer. Methods Eng. 36, (1993) 627}653. [19] M. Ganapathi, O. Polit, M. Touratier, Shear bending and torsion modelling for multilayered beams of rectangular cross-section. Proceedings of the Third International Conference of Composite Engineering (ICCE/3), New Orleans, 21}26 July, 1996, pp. 853}854. [20] R.M. Jones, Mechanics of Composite Materials, McGraw-Hill New York, 1975.
Shear #exible curved spline beam element for static analysis M. Ganapathi *, B.P. Patel , J. Saravanan, M. Touratier Institute of Armament Technology, Girinagar, Pune-411025, India Centre for Aeronautics System Studies and Analyses, Bangalore-560 003, India Laboratoire de Modelisation et Mecanique des Structures,URA-1776, CNRS-UPMC Paris 6-ENSAM-ENS Cachan,151, Bd de l'Hopital-75013 Paris, France
Abstract In this paper, an e$cient curved cubic B-spline beam element is developed based on "eld 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 e!ects. The element is based on laminated beam theory, which satis"es 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-"eld interpolations in their constrained physical limits causes poor convergence and unacceptable results due to locking. Hence, numerical experimentation is conducted to check these de"ciencies with a series of assumed shear/membrane strain functions, redistributed in a "eld 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. 1999 Elsevier Science B.V. All rights reserved. Keywords: Spline element; Field consistency; Curved beams; Laminates; Locking; Static
1. 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 o!er distinct advantages such as computational e$ciency, #exibility to model di!erent boundary conditions, good accuracy and convergence characteristics, and versatility, etc. Among the spline
* Corresponding author. Fax: #00-91-212-592139. E-mail address: [email protected] (M. Ganapathi) 0168-874X/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 8 7 4 X ( 9 9 ) 0 0 0 1 3 - X
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functions, the functions based on B-spline basis, which can have di!erent order of polynomial, is more in usage for the structural analysis. The study of static and dynamic behavior of various structural elements, using di!erent 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 "nite strip method whereas "nite 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 "nite point method by Zhou and Li [5]. Furthermore, "nite 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 e!ect of shear deformation. However, the application of spline function in conjunction with shear deformation theory has been sparsely treated in the literature [8}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}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 signi"cantly 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 di!erent order of spline functions for the constrained "eld variables [10], by constructing B-spline displacement "eld based on the inspection of the Timoshenko beam mode functions [12], and by introducing modi"ed shear modulii [8,9]. The shear locking behavior in the "nite element analysis of beams, plates and shells has been studied by many authors [13}17] in the context of elements based on conventional polynomial functions for the "eld variables. The more versatile approach, among the available techniques for alleviating locking such as reduced/selective integration scheme and assumed strain "elds, etc., is the "eld-consistent formulation [16,17]. It involves systematically eliminating spurious constraints causing shear/membrane locking in shear #exible "nite 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 "eld-consistent shear#exible 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 "eld consistency to "nd 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.
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2. 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, uI and w at point (x, z) from the median surface are expressed as functions of mid-plane displacement u and w and independent rotation h of the normal in xz plane, as uI(x, z)"u(x) (1#z/R)!zw V(x)#[f (z)#gI(z)] (w V(x)#h(x)), w(x, z)"w(x).
(1)
The functions f (z) and gI(z) are de"ned as f (z)"t/p sin(pz/t)!t/pb cos(pz/t),
(2a)
gI(z)"aIz#bI,
(2b)
where t is the thickness of the laminate. In Eqs. (2a), (2b) coe$cients bI are determined such that displacement component uI is continuous at the interface of the adjacent layers and zero at the mid point of the cross section. Finally, the coe$cients b and aI in Eqs. (2a), (2b) are computed from the requirement that the transverse shear stress pIVX is continuous at the interface of the adjacent layers and vanish at the top and bottom surfaces of the beam. The details of the derivations of constants, b, aI and bI can be found in Refs. [18,19]. The strains in the kth layer, +e,I are written as
e I e I # . 0 e
+e,I"
(3)
The mid-plane strain +e,, bending strain +e, and shear strain +e, in Eq. (3) are written as +e,I"u V#w/R, +e,I"!zw VV#[ f (z)#gI (z)] (w VV#h V)#zu V /R, +e,I"( f X#gI X) (w V#h),
(4a) (4b) (4c)
where the subscript comma denotes the partial derivative with respect to spatial coordinate succeeding it. For a composite laminated beam of layers of thickness tI(k"1, 2, 32), and the ply-angle jI(k"1, 2, 32), the necessary expressions for computing the sti!ness coe$cients, available in the literature [20], are used. The stress}strain relation for kth layer is written as
Q Q I +e,I, Q Q where QGH(i, j"1, 6) are the reduced sti!ness coe$cients of kth layer. The potential energy function ; for a curved beam with length l is given as +p,I"
RI> J (6) [+p,I]2 +e,I b dz dx ! +u w h, + f f m,2 dx, V U I RI where d is the vector of the degrees of freedom associated with the displacement "eld in a "nite element discretization and b is the width of the beam; f , and f are the forces per unit length in V U x and z directions, respectively; m is the moment per unit length of the beam. ;(d)"(1/2)
J
(5)
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3. Element description 3.1. 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 "eld variables u , w and h is the cubic B-spline of equal section length (h), and is given as (Fig. 1) O> O> u " a u, w" b u, G G G G G\ G\
O> h" c u, G G G\
Fig. 1. (a) Typical cubic B-spline. (b) Basis of cubic B-spline expression.
(7)
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in which each local cubic B-spline u has non-zero values over four consecutive sections with the G section-knot x"x as the centre, and is de"ned as G 0, x(x , G\ (x!x ), x )x)x , G\ G\ G\ h#3h(x!x )#3h(x!x )!3(x!x ), x )x)x , 1 G\ G\ G\ G\ G u" (8a) G 6h h#3h(x !x)#3h(x !x)!3(x !x), x )x)x , G> G> G> G G> (x !x), x )x)x , G> G> G> 0, x (x, G> 0, x(x , G\ 3(x!x ), x )x)x , G\ G\ G\ 3h#6h(x!x )!9(x!x ), x )x)x , 1 G\ G\ G\ G u " (8b) G V 6h !3h!6h(x !x)#9(x !x), x )x)x , G> G> G G> !3(x !x), x )x)x , G> G> G> 0, x (x G> a , b , c are the spline parameters. The strain}displacement as well as the stress}strain relationships G G G can be rewritten in terms of spline parameters, after substituting Eq. (7) into Eqs. (4a)}(5). Finally, the variation of the total potential energy given in Eq. (6) leads to the governing equation for the static analysis of the beam as
[[K ]#[K ]#[K ]]+d,"+F,, (9)
where [K ], [K ], [K ] are sti!ness matrices, concerning with membrane, bending and shear,
respectively; and +F, is the consistent load vector. These matrices are evaluated with respect to +d,, where +d, is the generalized vector based on spline parameters only. The displacements u and w, and rotation h at the ends of the spline beam element can be expressed in terms of the spline parameters as 1 1 u " (a #4a #a ), u " (!a #a ); M V 2h \ M 6 \ 1 1 u " (a #4a #a ), u " (!a #a ); O O> MO V 2h O\ O> MO 6 O\ 1 1 w " (b #4b #b ), w " (!b #b ); 6 \ V 2h \ 1 1 w " (b #4b #b ), w " (!b #b ); O 6 O\ O O> O V 2h O\ O> 1 1 h " (c #4c #c ), h " (!c #c ); 6 \ V 2h \ 1 1 h " (c #4c #c ), h " (!c #c ). O 6 O\ O O> O V 2h O\ O>
(10)
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If the element is divided into q equal sections, there will be, all together, [3;(q#3)] spline parameters to de"ne the displacements (u , w) and rotation (h) functions. Instead of the spline parameters a , a , b , b , c and c at the ends of the element and, a , a , b , O O O \ O> \ b , c and c outside the element, one can use the displacements, rotation and their derivaO> \ O> tives at the ends, i.e. u , u , w, w, , h and h, . Thus, within the element, for the present case, there V V V are [3;(q!1)] interior spline parameters which are included in the generalized vector +d,, after the transformation based on (10). In this way, the original generalized vector of spline parameters +d, will now consist of physical coordinates vector (displacements and their derivatives) pertaining to the ends of the element and vector of interior spline parameters. In view of the modi"cation in the original vector of spline parameters, mass and sti!ness matrices shown in (9) are to be updated with respect to the modi"ed generalized vector. This procedure enables us to assemble the element level matrices and introduce the desired boundary conditions, as followed in the standard "nite element procedure. Here, we shall experiment with series of assumed membrane/shear strain functions for a displacement-type cubic B-spline element to derive the optimal membrane/shear strain de"nitions that leave the element free of all problems for most of the boundary conditions. The systematic procedure adopted here, as per the "eld consistency principle, is by establishing the consistency of membrane/shear strain "elds by simply smoothing the w and h to the order of the functions for u and w (Level 1 functions below-the Q-element), by smoothing membrane/shear strain "elds to M V V linear functions (Level 2 functions-the L-element), and by smoothing these strain functions to constant form (Level 3 functions-the C-element). 3.2. O-element } The original element ( xeld-inconsistent element } ISIM) In a conventional approach, the interpolating functions u based on B-spline basis, as de"ned by G Eq. (8a), are cubic and associate four constants within intervals, and are used for de"ning the "eld variables interpolations, see (7). Thus, the membrane and shear strains can be derived from
0,
x(x
, G\ a #a x#a x#a x, x )x)x , G\ G\ a #a x#a x#a x, x )x)x , G\ G u" G a #a x#a x#a x, x )x)x , G G> a #a x#a x#a x, x )x)x , G> G> 0, x (x, G> 0,
(11a)
x(x
, G\ b #b x#b x#b x, x )x)x , G\ G\ b #b x#b x#b x, x )x)x , G\ G w" G b #b x#b x#b x, x )x)x , G G> b #b x#b x#b x, x )x)x , G> G> 0, x (x, G>
(11b)
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x(x
, G\ c #c x#c x#c x, x )x)x , G\ G\ c #c x#c x#c x, x )x)x , G\ G h" G c #c x#c x#c x, x )x)x , G G> c #c x#c x#c x, x )x)x , G> G> 0, x (x. G> as
187
0,
(11c)
x(x
0,
, G\ (a #b /R)#(2a #b /R)x#(3a #b /R)x#(b /R)x, x )x)x , G\ G\ (a #b /R)#(2a #b /R)x#(3a #b /R)x#(b /R)x, x )x)x , G G> eN" G (a #b /R)#(2a #b /R)x#(3a #b /R)x#(b /R)x, x )x)x , G G> (a #b /R)#(2a #b /R)x#(3a #b /R)x#(b /R)x, x )x)x , G> G> 0, x (x, G> (12a)
x(x , G\ (b #c )#(2b #c )x#(3b #c )x#(c )x, x )x)x , G\ G\ (b #c )#(2b #c )x#(3b #c )x#(c )x, x )x)x , G\ G eQ"( f, #g,I) G X X (b #c )#(2b #c )x#(3b #c )x#(c )x, x )x)x , G G> (b #c )#(2b #c )x#(3b #c )x#(c )x, x )x)x , G> G> 0, x (x. G> (12b) 0,
If the element is to model a situation in which in-extensional bending and the Kirchho! limit of classical thin beam predominate, the requirements that the membrane and shear strain should vanish, produce in e!ect the following conditions: Membrane case: (a #b /R)"0, (a #b /R)"0, (a #b /R)"0, (a #b /R)"0,
(2a #b /R)"0, (2a #b /R)"0, (2a #b /R)"0, (2a #b /R)"0,
(3a #b /R)"0, (3a #b /R)"0, (3a #b /R)"0, (3a #b /R)"0,
(b /R)"0, (b /R)"0, (b /R)"0, (b /R)"0,
x )x)x , G\ G\ x )x)x , G\ G x )x)x , G G> x )x)x . G> G> (13a)
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Shear case: (b #c )"0, (b #c )"0, (b #c )"0, (b #c )"0,
(2b #c )"0, (2b #c )"0, (2b #c )"0, (2b #c )"0,
(3b #c )"0, (3b #c )"0, (3b #c )"0, (3b #c )"0,
(c )"0, (c )"0, (c )"0, (c )"0,
x
)x)x , G\ G\ x )x)x , G\ G x )x)x , G G> x )x)x. G>
(13b)
Using the element, based on the original interpolation functions employing directly in deriving the constrained strain "elds, our studies here show that this element locks (i.e. spurious over-sti!ening that increases inde"nitely with reduction in beam thickness). As per "eld consistency paradigm, the derivative functions for u and w do not match that for the w and h in the membrane and shear V V strain de"nitions in precisely these cubic terms. Field consistency requires, in a simple sense, that the membrane/shear strain must be interpolated in a consistent manner. Thus, w term in e, and h in e must be consistent with the "eld functions for u and w , respectively. M V V 3.3. Q-element } the element based on Level 1 consistency (CMCS-1) At the simplest level of "eld-consistency, we consider the use of "eld redistributed substitute interpolation functions which include only those speci"c terms that must be made "eld-consistent, as outlined in Refs. [16,17]. This is achieved here by smoothing the original interpolation function in a least-squares-accurate fashion to the desired form, i.e. the functions that are consistent with the derivative functions. Here, we need smoothed functions for w and h which are consistent with the interpolations u , and w for the use in the membrane and shear strains de"nitions, respectively. It M V V can be noted that u and w are of the quadratic form, as seen from (8b) in de"ning the function M V V u which is involved in evaluating the derivative of the "eld variables, as per O-level description. G V This means that the smoothed functions we must derive for w, and h are, designated as u, obtained G by smoothing the original functions u to be a least-squares form consistent with the derivative G functions u . This operation is simple and the substitute interpolation functions obtained, in this G V way, are given as follows:
0,
x(x
, G\ 1/120!1/10(x!x )/h#(1/4) (x!x )/h, x )x)x , G\ G\ G\ G\ 17/120#(4/5) (x!x )/h!(x!x )/4h, x )x)x , G\ G\ G\ G u" G 83/120!(36/120) (h!x #x)/h!(30/120) (h!x #x)/h, x )x)x , G> G> G G> 19/120!(2/5) (h!x #x)/h#(1/4) (h!x #x)/h, x )x)x , G> G> G> G> 0, x (x. G> (14) The functions u is retained for u and w in the membrane and shear strains computation. G V M V V However, we shall see below that this element can still lock for certain sets of boundary conditions, even though consistent de"nitions for the membrane and shear strain "elds have been assured within the element domain. This suggests that an inconsistency in the shear/membrane strain
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189
de"nitions have been introduced for some set of boundary conditions. It is therefore necessary to examine lower levels of consistency based on further sets of assumed membrane/shear strain functions to see if these locking mechanisms/de"ciency can be removed.
3.4. L-element } the element based on Level 2 consistency (CMCS-2) It is logical now to consider smoothed interpolation functions based on linear form for w and h pertaining to membrane and shear strains, respectively. This will necessitate that the derivative functions de"ning u and w will also have to be smoothed to linear form. The shape functions M V V required here can be denoted u for w and h, and u for u and w , respectively. The G G V M V V interpolation functions u are obtained by smoothing the original derivative forms u in G V G V a least-squares-accurate operation over the element domain. These functions are then
x(x , G\ !1/30#(3/20) (x!x )/h, x )x)x , G\ G\ G\ 11/60#(11/20) (x!x )/h, x )x)x , G\ G\ G u" G 11/15!(11/20) (h!x #x)/h, x )x)x , G> G G> 7/60!(3/20) (h!x #x)/h, x )x)x , G> G> G> 0, x (x. G>
(15a)
x(x , G\ !1/(12h)#(1/2h) (x!x )/h, x )x)x , G\ G\ G\ 3/4h!(1/2h) (x!x )/h, x )x)x , G\ G\ G u " G V !1/4h!(1/2h) (h!x #x)/h, x )x)x , G> G G> !5/12h!(1/2h) (h!x #x)/h, x )x)x , G> G> G> 0, x (x. G>
(15b)
0,
0,
Numerical experiments below show that this element is identical in behavior as seen in the case of Level 1, and it locks for most of the practical boundary conditions for a beam. Thus smoothing to level 2 has failed to remove the locking mechanism.
3.5. C-element } the element based on Level 3 consistency (CMCS-3) We now go down to the lowest level (a constant form for representing the w and h) and correspondingly, the derivative functions de"ning u , and w will also have to be smoothed to M V V constant form. The interpolation functions for this case is assigned as u for w and h, and u for G G V u and w , respectively. The functions u are obtained by smoothing the original derivative M V V G V forms u in a least-squares accurate operation over the element domain. It can be shown that G V
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these functions to be used are
x(x , G\ 1/24, x )x)x , G\ G\ 11/24, x )x)x , G\ G u" G 11/24, x )x)x , G G> 1/24, x )x)x , G> G> 0, x (x, G> 0,
0,
(16a)
x(x
, G\ 1/6h, x )x)x , G\ G\ 1/2h, x )x)x , G\ G u " G V !1/2h, x )x)x , G G> !1/6h, x )x)x , G> G> 0, x (x. G>
(16b)
The numerical analysis below shows that this element does not lock for the boundary conditions considered here, i.e. its accuracy is insensitive to a very large variation in the thickness parameter of the beam. It can be noted that this can be achieved by using a reduced integration rule, i.e. a Gaussian one-point integration of membrane and shear energy will produce the same results. However, for the thick beam, the numerical study given below brings out di!erent results compared to the actual one. The procedure adopted by "eld consistency approach permits greater #exibility in the choice of integration order for the energy terms, as these can now be determined by the other considerations (such variable thickness, more accurate integration of 1/R terms, etc.) and not dictated by the need for "eld consistency.
4. 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 di!erent functions for the redistributed strain "elds through the "eld consistent strategy. 4.1. Test case 1: clamped}clamped isotropic circular arc with sector angle "1803 The numerical results for the static analyses are evaluated using the element developed in the earlier section and are shown in Table 1. The results obtained here are the maximum de#ection of the arc having slenderness ratio of ¸/t"10, 100 and 10. They are also in the form of convergence studies for de#ections (w "w [EI/PR], where P is the load at the centre of the arc, and EI is the
M. Ganapathi et al. / Finite Elements in Analysis and Design 32 (1999) 181}202
191
Table 1 Non-dimensional de#ection at the centre of clamped}clamped circular arc with sector angle 1803 with di!erent consistent strategies (M: Membrane, S: Shear, O: Original, Q: Quadratic, L: Linear, C: Constant, RI: Reduced Integration, CL: Classical Theory) q ¸/t 10
100
10
MO
MQ
ML
MC
MO
MQ
ML
MC
MO
MQ
ML
MC
4 SO SQ SL SC RI
0.01916 0.01923 0.02003 0.02050 0.11571
0.01921 0.01929 0.02009 0.02057
0.01993 0.02001 0.02089 0.02142
0.02106 0.02116 0.02229 0.02296
0.00033 0.00034 0.00037 0.00058 0.01264
0.00033 0.00034 0.00037 0.00059
0.00036 0.00037 0.00040 0.00068
0.00050 0.00054 0.00064 0.00806
0.00000 0.00000 0.00000 0.00000 0.01123
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00790
6 SO SQ SL SC RI
0.02506 0.02508 0.02549 0.02584 0.10069
0.02508 0.02509 0.02550 0.02585
0.02517 0.02518 0.02559 0.02595
0.02556 0.02558 0.02601 0.02635
0.00393 0.00411 0.00435 0.00615 0.01163
0.00402 0.00421 0.00446 0.00637
0.00426 0.00448 0.00476 0.00694
0.00538 0.00575 0.00623 0.01085
0.00000 0.00000 0.00000 0.00000 0.01048
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.01068
8 SO SQ SL SC RI
0.02595 0.02595 0.02622 0.02679 0.10479
0.02595 0.02595 0.02622 0.02679
0.02602 0.02602 0.02629 0.02686
0.02624 0.02625 0.02652 0.02709
0.00913 0.00926 0.00945 0.01046 0.01244
0.00921 0.00934 0.00954 0.01055
0.00932 0.00945 0.00965 0.01070
0.00974 0.00989 0.01011 0.01143
0.00000 0.00000 0.00000 0.00000 0.01127
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.01126
10 SO SQ SL SC RI
0.02628 0.02628 0.02644 0.02704 0.10014
0.02628 0.02629 0.02645 0.02704
0.02632 0.02632 0.02648 0.02708
0.02652 0.02652 0.02668 0.02728
0.01089 0.01093 0.01101 0.01139 0.01250
0.01091 0.01096 0.01104 0.01142
0.01094 0.01099 0.01107 0.01146
0.01105 0.01110 0.01118 0.01163
0.00000 0.00000 0.00000 0.00000 0.01140
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.01147
14 SO SQ SL SC RI
0.02660 0.02660 0.02669 0.02718 0.09752
0.02660 0.02660 0.02669 0.02718
0.02662 0.02662 0.02671 0.02721
0.02675 0.02675 0.02685 0.02734
0.01162 0.01163 0.01166 0.01174 0.01263
0.01162 0.01163 0.01166 0.01174
0.01163 0.01164 0.01166 0.01174
0.01164 0.01165 0.01167 0.01176
0.00000 0.00000 0.00000 0.00000 0.01157
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.01159
16 SO SQ SL SC RI
0.02668 0.02668 0.02675 0.02720 0.09669
0.02668 0.02668 0.02675 0.02720
0.02670 0.02670 0.02677 0.02722
0.02682 0.02682 0.02689 0.02733
0.01171 0.01172 0.01173 0.01178 0.01265
0.01171 0.01172 0.01173 0.01178
0.01171 0.01172 0.01173 0.01178
0.01172 0.01172 0.01174 0.01178
0.00000 0.00000 0.00000 0.00000 0.01160
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.01162
20 SO SQ SL SC RI
0.02679 0.02679 0.02684 0.02720 0.09423
0.02679 0.02679 0.02684 0.02720
0.02680 0.02680 0.02685 0.02721
0.02690 0.02690 0.02694 0.02731
0.01178 0.01178 0.01179 0.01180 0.01265
0.01178 0.01178 0.01179 0.01180
0.01178 0.01178 0.01179 0.01180
0.01178 0.01179 0.01179 0.01181
0.00000 0.00000 0.00000 0.00000 0.01163
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.01164
24 SO SQ SL SC RI
0.02686 0.02686 0.02689 0.02719 0.09240
0.02686 0.02686 0.02689 0.02719
0.02687 0.02687 0.02690 0.02720
0.02695 0.02695 0.02698 0.02728
0.01180 0.01181 0.01181 0.01181 0.01264
0.01180 0.01181 0.01181 0.01181
0.01181 0.01181 0.01181 0.01182
0.01181 0.01181 0.01181 0.01182
0.00000 0.00000 0.00000 0.00000 0.01164
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.01165
CL 0.01166
0.01166
0.01166
192
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#exural rigidity) with the number of B-spline sections q. The solutions obtained based on the original functions for the "eld variables (u, w, h), which results in "eld inconsistency approach for the constrained strains (membrane and shear), and also employing di!erent levels "eld consistent representation for the constrained strain functions (quadratic, linear, constant) are indicated in the table. The comparative results quoted in the table are calculated using one point numerical reduced integration scheme (RI) for the evaluation of membrane and shear energy terms considering the original functions, and the analytical solutions within the con"nes of Euler}Bernoulli beam theory (CL). It can be noted here that the numerical experiments with various combinations of reduced integration techniques for estimating the membrane and shear energy terms were conducted. The best accuracy was attained for one-point quadrature scheme for both membrane and shear energy terms. For the sake of brevity, the detailed study is not presented here. It is very clear from Table 1 that the rate of convergence is very good for both thick and thin beams with the increase in the spline sections. It may be inferred from Table 1 that for fairly thick and thin cases (¸/t"10, 100), the results obtained adopting "eld inconsistency manner are in close agreement with those corresponding to the di!erent levels of "eld consistent strategy for the constrained strain functions. However, the reduced integration scheme yields erroneous results for fairly thick beam ¸/t"10, but it gives acceptable results for fairly as well as extremely thin situations as highlighted in Table 1. Furthermore, it is amply clear from the table that, with the reasonable number of spline sections q for modeling the very thin beam (¸/t"10) adequately, the element exhibits severe locking when the full integration scheme is employed. With lowering the order of level for the "eld consistency requirement in de"ning the constrained strain terms from quadratic to linear function does not alleviate the locking. It may be inferred from the table that the consistent way of representing the constrained strains (membrane and shear) at constant level produces remarkable recovery from the locking syndrome. However, applying the consistent strategy either for shear or membrane alone cannot free the spurious energy locked in the element. The numerical experiment conducted here, in general, suggests that for the beam thickness parameter ¸/t of the order 100, the original spline element does not seem to su!er much from locking problems whereas it is susceptible to membrane and shear locking for the extreme thin case. Furthermore, this study again reveals that unlike element employing the reduced integration scheme, which may not be suitable for the problems wherein a higher order of integration is needed (for instance tapered beams), consistent element frees the order of integration and thus, is applicable for both thick and extreme thin situations, as expected [16,17] The results obtained using the consistent formulation for the extremely thin beam agree very well with the classical solutions. The stress distributions obtained with inconsistency approach, and the consistency formulation with constant functions for the constrained strains are given in Fig. 2 (q"24, ¸/t"10). For the inconsistency approach (IMIS), one can notice the completely erroneous results for axial thrust, bending moment and shear stress, i.e. large values for axial and shear stresses, and very small values for moment are the typical of a locking solution. But, the stresses evaluated based on consistent formulation (CMCS-3) yield accurate distributions and match well with the analytical results. 4.2. Test case 2: clamped-free isotropic circular arc with sector angle "1803 A clamped-free circular arc loaded at the free end is considered here. De#ections obtained at the free end (w "w [EI/PR]) are given in Table 2 by varying the aspect ratio of the beam from fairly D
M. Ganapathi et al. / Finite Elements in Analysis and Design 32 (1999) 181}202
193
Fig. 2. Stress and moment distributions for clamped}clamped isotropic are using consistent and inconsistent models: (a) axial stress, (b) bending moment, (c) shear stress.
thick one to extreme thin situation. The di!erent results calculated considering full and reduced integration schemes are highlighted along with those of the "eld consistent formulation at di!erent levels of strategies for the membrane and shear strain "elds. Like test case 1, one can infer from Table 2 (extreme thin) that the element exhibits signi"cant locking, while performing the full
194
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Table 2 Non-dimensional de#ection at free end of clamped-free circular arc with sector angle 1803 with di!erent consistent strategies (M: Membrane, S: Shear, O: Original, Q: Quadratic, L: Linear, C: Constant, RI: Reduced Integration, CL: Classical Theroy) q ¸/t 10
100
10
MO
MQ
ML
MC
MO
MQ
ML
MC
MO
MQ
ML
MC
2 SO SQ SL SC RI
1.41439 1.41801 1.43074 1.76082 11.7550
1.43920 1.44309 1.45318 1.76278
1.54838 1.55680 1.56201 1.79268
2.27168 2.44178 4.23596 23.5202
0.89325 0.89397 0.90173 1.68494 11.5980
0.92233 0.92375 0.93361 1.68495
1.12957 1.21052 1.47907 1.68853
1.32338 1.34737 3.18143 23.4813
0.84634 0.84657 0.84647 1.68492 11.4935
0.84641 0.84642 0.84628 1.68536
0.84609 0.84607 1.47727 1.68749
1.29180 1.29228 3.17112 23.7800
4 SO SQ SL SC RI
1.61428 1.61447 1.61587 1.61787 2.14890
1.61550 1.61569 1.61712 1.61897
1.61644 1.61663 1.61810 1.61985
1.76314 1.77625 1.94809 3.65888
1.34129 1.34562 1.34830 1.37467 2.00516
1.38385 1.38977 1.39337 1.40769
1.42978 1.44312 1.46463 1.52218
1.55748 1.57900 1.64886 3.60176
0.84629 0.84634 0.84630 1.12992 2.00334
0.84648 0.84648 0.84622 1.12973
0.84626 0.84614 0.85679 1.50110
1.28457 1.28543 1.63586 3.62173
6 SO SQ SL SC RI
1.62194 1.62196 1.62243 1.62420 1.82209
1.62204 1.62207 1.62254 1.62431
1.62209 1.62211 1.62258 1.62436
1.68624 1.69163 1.76199 2.17560
1.53418 1.53540 1.53582 1.53963 1.67697
1.54221 1.54348 1.54388 1.54817
1.54475 1.54606 1.54642 1.55451
1.57657 1.57859 1.58588 2.11737
0.84699 0.84646 0.84691 1.01513 1.67582
0.84662 0.84649 0.84652 1.01507
0.84592 0.84599 1.30807 1.43537
1.28082 1.28044 1.46295 2.11221
8 SO SQ SL SC RI
1.62278 1.62279 1.62307 1.62449 1.75945
1.62280 1.62281 1.62309 1.62451
1.62281 1.62282 1.62309 1.62452
1.66182 1.66492 1.70200 1.85457
1.56458 1.56486 1.56498 1.56567 1.61355
1.56613 1.56641 1.56652 1.56728
1.56639 1.56677 1.56678 1.56780
1.57498 1.57546 1.57909 1.79701
0.84590 0.84570 0.84581 0.96485 1.61223
0.84584 0.84577 0.84623 0.96494
0.84530 0.84533 1.22990 1.34657
1.27988 1.27998 1.39740 1.79728
10 SO SQ SL SC RI
1.62309 1.62310 1.62329 1.62444 1.73921
1.62310 1.62310 1.62330 1.62445
1.62310 1.62310 1.62330 1.62445
1.64989 1.65185 1.67328 1.74272
1.56960 1.56968 1.56971 1.56987 1.59285
1.57000 1.57008 1.57012 1.57029
1.57005 1.57013 1.57016 1.57036
1.57360 1.57381 1.57612 1.68614
0.84625 0.84666 0.84626 0.93882 1.59135
0.84647 0.84677 0.84647 0.93853
0.84559 0.84716 1.16963 1.27676
1.27844 1.27752 1.36217 1.68607
14 SO SQ SL SC RI
1.62339 1.62339 1.62351 1.62431 1.72651
1.62339 1.62339 1.62351 1.62431
1.62339 1.62339 1.62351 1.62431
1.63826 1.63916 1.64743 1.66817
1.57110 1.57111 1.57112 1.57114 1.57966
1.57116 1.57117 1.57117 1.57119
1.57116 1.57117 1.57118 1.57120
1.57239 1.57247 1.57365 1.61298
0.84691 0.84638 0.84601 0.91001 1.57622
0.84652 0.84634 0.84670 0.90996
0.84567 0.84585 1.08779 1.17342
1.28851 1.28718 1.34086 1.62034
16 SO SQ SL SC RI
1.62348 1.62348 1.62357 1.62426 1.72422
1.62348 1.62348 1.62357 1.62426
1.62348 1.62348 1.62357 1.62426
1.63507 1.63571 1.64111 1.65395
1.57123 1.57123 1.57124 1.57125 1.57721
1.57125 1.57126 1.57126 1.57127
1.57125 1.57126 1.57126 1.59127
1.57211 1.57218 1.57308 1.59920
0.84263 0.84267 0.84277 0.89857 1.57459
0.84278 0.84296 0.84414 0.89816
0.84480 0.84302 1.06083 1.13468
1.26197 1.26467 1.31036 1.59353
20 SO SQ SL SC RI
1.62359 1.62359 1.62366 1.62419 1.72201
1.62359 1.62359 1.62366 1.62419
1.62359 1.62359 1.62366 1.62419
1.63105 1.63139 1.63389 1.63973
1.57130 1.57130 1.57131 1.57131 1.57481
1.57131 1.57131 1.57131 1.57132
1.57131 1.57131 1.57131 1.57132
1.57181 1.57185 1.57243 1.58559
0.84680 0.84639 0.84607 0.88993 1.57408
0.84669 0.84814 0.84662 0.88856
0.84800 0.84825 1.02462 1.08816
1.27178 1.28896 1.30153 1.57513
CL 1.570796
1.570796
1.570796
M. Ganapathi et al. / Finite Elements in Analysis and Design 32 (1999) 181}202
195
Fig. 3. Stress and moment distributions for clamped-free isotropic arc using consistent and inconsistent models: (a) axial stress, (b) bending moment, (c) shear stress.
integration scheme for evaluating the sti!ness matrices. By lowering the order of level for the "eld consistency requirement in de"ning the constrained strain terms from quadratic to linear function, locking to some extent is getting freed but the spurious energy is completely removed for the constrained strains with constant functions. The qualitative trend of the results obtained using "eld
196
M. Ganapathi et al. / Finite Elements in Analysis and Design 32 (1999) 181}202
inconsistency as well as consistency manner is in general, similar to that of Test case 1. The results based on "eld consistency manner once again amply demonstrate that it is in very good agreement with those of the original functions adopting full integration scheme for the thick situation and they match well with the classical solution for the thin cases. The stress oscillations due to inconsistency formulation, and the actual stress distributions obtained using the consistency formulation with constant functions for the constrained strains are given in Fig. 3 (q"24, ¸/t"10). For the inconsistency approach (IMIS), one can notice the stress oscillation, around the free end at which load is applied, and erroneous results (also large values for axial and shear stresses are the typical of a locking solution). But, the stresses evaluated using the consistent formulation (CMCS-3) match well with the analytical results. 4.3. Test case 3: pinched circular ring The static analysis of isotropic pinched ring is carried out using half-ring model idealization. The variation of the de#ection at the loading point w "2w [EI/PR] with the aspect ratio, spline section and di!erent consistent strategies is described in Table 3. The element behavior for this problem, in general, is quite similar to those of Test case 1. However, unlike the previous test cases, it is apparently visible from Table 3 that the results evaluated for thin ring adopting reduced integration scheme is di!erent from the classical solutions and it seems to decrease with the increase in the spline sections. But, the "eld-consistent element based on constant level approximation alone for the spurious energy terms behaves very well for very thin beam and the results are in very good agreement with Euler}Bernoulli analysis. Fig. 4 (q"24, ¸/t"10) shows the stress oscillations, near the symmetric boundary conditions (also large values for axial and shear stresses, and very small values for moment are the typical of a locking solution), due to the inconsistency formulation. The stress distributions evaluated using the consistency formulation (CSCM-3) match very well with the existing solutions. Like, the previous examples, the consistent formulation ensures the accurate results. 4.4. Test case 4: analysis of simply supported composite laminates with sinusoidal load [P sin(px/l)] To see e$cacy of the element formulation, the problems of straight cross-ply (03, 03/903, 03/903/03) laminated beams, and symmetric cross-ply curved laminates (03/903/03) are considered using number of spline sections q"24. The material properties used here are E /E "25, G /E "0.5, G /E "0.2, G /E "0.5, l "0.25. The value of the displacement for the straight beam case is plotted against the aspect ratio in Fig. 5. The formulation based on CSCM-3 alone yields accurate results and compare very well with exact solution. The variation of stresses through the thickness of the laminate is shown in Fig. 6 along with the exact solution. They are found to be in good agreement. Similar study is made for the curved cross-ply composite laminates (03/903/03), having sector angle 603 and subjected to sinusoidal load, for di!erent R/t. Table 4 gives displacement and stresses obtained from the present formulation and compares with elasticity solution.
M. Ganapathi et al. / Finite Elements in Analysis and Design 32 (1999) 181}202
197
Table 3 Non-dimensional de#ection (= "2w EI/PR) at the loading point of pinched ring with di!erent consistent strategies (M: M Q: Quadratic, L: Linear, C: Constant, RI: Reduced Integration, CL: Classical Theroy) Membrane, S: Shear, O: Original, q ¸/t 10
100
10
MO
MQ
ML
MC
MO
MQ
ML
MC
MO
MQ
ML
MC
2 SO SQ SL SC RI
0.15034 0.15047 0.15087 0.15112 434.604
0.15187 0.15201 0.15242 0.15268
0.15370 0.15387 0.15430 0.15465
0.23684 0.25872 0.25951 3.36113
0.05242 0.05245 0.05245 0.05257 434.485
0.08130 0.08133 0.08133 0.08152
0.14274 0.14293 0.14293 0.14789
0.17402 0.17426 0.17427 3.35486
0.00000 0.00000 0.00000 0.00000 461.004
0.00000 0.00000 0.00000 0.00000
0.14217 0.14216 0.14214 0.14781
0.17324 0.17325 0.17329 2.96077
4 SO SQ SL SC RI
0.15508 0.15509 0.15519 0.15538 121.319
0.15511 0.15512 0.15521 0.15541
0.15512 0.15512 0.15522 0.15542
0.16516 0.16566 0.16652 0.42411
0.14146 0.14162 0.14163 0.14209 121.025
0.14362 0.14379 0.14380 0.14427
0.14411 0.14430 0.14430 0.14493
0.15193 0.15225 0.15227 0.41751
0.00000 0.00000 0.00000 0.00000 115.192
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.12479
0.14549 0.14550 0.14550 0.39485
6 SO SQ SL SC RI
0.15524 0.15524 0.15529 0.15547 76.9276
0.15524 0.15524 0.15529 0.15547
0.15524 0.15524 0.15529 0.15547
0.15958 0.15977 0.16007 0.22198
0.14809 0.14812 0.14813 0.14821 76.4563
0.14829 0.14832 0.14832 0.14840
0.14831 0.14834 0.14834 0.14843
0.14979 0.14984 0.14986 0.21514
0.00000 0.00000 0.00000 0.00000 72.0788
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.09689
0.14312 0.14311 0.14310 0.21663
8 SO SQ SL SC RI
0.15529 0.15529 0.15532 0.15547 57.7407
0.15529 0.15529 0.15532 0.15547
0.15529 0.15529 0.15533 0.15547
0.15788 0.15800 0.15817 0.18152
0.14871 0.14872 0.14872 0.14873 57.1306
0.14874 0.14875 0.14875 0.14877
0.14875 0.14875 0.14875 0.14877
0.14929 0.14931 0.14932 0.17457
0.00000 0.00000 0.00000 0.00000 60.1938
0.00000 0.00000 0.00000 0.00000
0.00000 0.00000 0.00000 0.07653
0.14252 0.14256 0.14260 0.17297
10 SO SQ SL SC RI
0.15532 0.15532 0.15535 0.15546 46.4402
0.15532 0.15532 0.15535 0.15546
0.15532 0.15532 0.15535 0.15547
0.15709 0.15716 0.15727 0.16868
0.14881 0.14881 0.14881 0.14882 45.7038
0.14882 0.14882 0.14882 0.14882
0.14882 0.14882 0.14882 0.14882
0.14909 0.14910 0.14911 0.16175
0.00001 0.00001 0.00001 0.00001 54.5013
0.00001 0.00001 0.00001 0.00001
0.00001 0.00001 0.00001 0.06260
0.14237 0.14235 0.14226 0.16165
14 SO SQ SL SC RI
0.15536 0.15536 0.15537 0.15546 33.6433
0.15536 0.15536 0.15537 0.15546
0.15536 0.15536 0.15537 0.15546
0.15635 0.15638 0.15643 0.16034
0.14884 0.14884 0.14884 0.14884 32.6745
0.14884 0.14884 0.14884 0.14884
0.14884 0.14884 0.14884 0.14884
0.14895 0.14895 0.14896 0.15352
0.00006 0.00006 0.00006 0.00006 30.4532
0.00008 0.00008 0.00008 0.00009
0.00008 0.00008 0.00008 0.04558
0.14278 0.14283 0.14295 0.15420
16 SO SQ SL SC RI
0.15537 0.15537 0.15538 0.15545 29.6797
0.15537 0.15537 0.15538 0.15545
0.15537 0.15537 0.15538 0.15545
0.15615 0.15617 0.15621 0.15875
0.14884 0.14884 0.14884 0.14884 28.6074
0.14884 0.14884 0.14884 0.14884
0.14884 0.14884 0.14884 0.14884
0.14892 0.14892 0.14893 0.15198
0.00013 0.00013 0.00013 0.00014 23.2414
0.00018 0.00018 0.00018 0.00020
0.00018 0.00018 0.00018 0.04013
0.14208 0.14201 0.14223 0.15185
20 SO SQ SL SC RI
0.15538 0.15538 0.15539 0.15545 24.16929
0.15538 0.15538 0.15539 0.15545
0.15538 0.15538 0.15539 0.15545
0.15589 0.15591 0.15593 0.15716
0.14884 0.14884 0.14884 0.14884 22.91724
0.14884 0.14884 0.14884 0.14884
0.14884 0.14884 0.14884 0.14884
0.14889 0.14889 0.14889 0.15045
0.00047 0.00048 0.00048 0.00053 28.17586
0.00068 0.00068 0.00069 0.00075
0.00069 0.00069 0.00069 0.03268
0.14227 0.14214 0.14171 0.14992
24 SO SQ SL SC RI
0.15539 0.15539 0.15539 0.15544 20.52279
0.15539 0.15539 0.15539 0.15544
0.15539 0.15539 0.15539 0.15544
0.15575 0.15576 0.15577 0.15645
0.14884 0.14884 0.14884 0.14885 19.12632
0.14884 0.14884 0.14884 0.14885
0.14884 0.14884 0.14884 0.14885
0.14887 0.14887 0.14888 0.14977
0.0014 0.00142 0.00142 0.00157 18.01243
0.002 0.00201 0.00201 0.00222
0.00201 0.00203 0.00203 0.02849
0.14234 0.14227 0.14171 0.14942
CL 0.148778
0.148778
0.148778
198
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Fig. 4. Stress and moment distributions for isotropic pinched circular ring using consistent and inconsistent models: (a) axial stress, (b) bending moment, (c) shear stress.
Fig. 7 presents a comparison for the variation of shear stress distribution along the thickness with those of elasticity solutions. The results are, in general, very good agreement with the analytical solution.
M. Ganapathi et al. / Finite Elements in Analysis and Design 32 (1999) 181}202
199
Fig. 5. Normalized transverse displacement with aspect ratio for cross-ply straight beam.
Fig. 6. Stress distribution in straight cross-ply beam: (a) normalized shear stress, (b) normalized in-plane stress.
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Fig. 7. Normalized transverse shear stress distribution in curved cross-ply (R/t"20, 50).
M. Ganapathi et al. / Finite Elements in Analysis and Design 32 (1999) 181}202
201
Table 4 Comparison of results with elasticity solution (ES) for cross-ply simply supported laminated arc ( "603) R/t
20 50 100 1000 10000
pH V
wH Present
ES
Present
ES
2.3767 2.0245 1.9690 1.9452 1.9442
2.38 2.02 1.96 } }
!8.3986 !7.9440 !7.8467 !7.7820 !7.7767
!8.5 !7.98 !7.87 } }
Note: wH"10 E w(¸/2, 0)t/p R pH"10 p (¸/2, !t/2)t/p R. * M V V M
5. Conclusions The shear-membrane locking phenomena in the shear #exible curved element based on cubic B-spline functions for the displacement "elds have been analyzed using the "eld-consistency approach. The "eld-consistent redistribution of membrane and shear strain "elds 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 e!ectiveness of the element has been demonstrated here for the static analysis of a wide range of problems without membrane or shear locking e!ects emerging.
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