General curved beam elements based on the assumed strain fields

0045-7949(94)00469-2

GENERAL

Compulers & Struc,ures Vol. 55, No. 3. pp. 379-386, 1995 Copyright Q 1995 Elsevter Scmce Ltd Printed in Great Bntain. All rights reserved 0045.7949195 $9.50 + 0 00

CURVED BEAM ELEMENTS BASED ASSUMED STRAIN FIELDS

ON THE

Jong-keun Choit and Jang-keun Lim$ tAutomotive IDepartment

Technology Laboratory, Institute for Advanced Engineering, C.P.O. Box 2849, Seoul, Korea of Mechanical Engineering, Hanyang University, 17 Haengdang-Dong, Sungdong-Gu, Seoul 133-791, Korea (Received

28 December

1993)

Abstract-Simple two-noded and three-noded general curved beam elements have been formulated on the basis of assumed strain fields and Timoshenko’s beam theory. The two-noded element is formulated from constant strain fields and the three-noded one is from linear strain fields. These curved beam elements, designed in a local curvilinear coordinate system, are transformed into a global Cartesian coordinate system in order to analyze effectively the general curved beam structures located arbitrarily in space. Since strain functions are assumed independently, these elements are free from any locking phenomena. Through various numerical tests, it is shown that the suggested general curved beam elements give better convergent characteristics than the modified isoparametric curved beam elements that have been shown in existing studies. These elements are also easy to formulate due to their consistent form of strain fields,

1. INTRODUCTION

two-noded element which has constant strain fields, the other is a three-noded element which has linear strain fields. These two elements, formulated in the local curvilinear coordinates, are transformed into a global Cartesian coordinate system in order to analyze effectively certain curved beam structures in space.

various curved beam structures effectively by the finite element method, curved beam elements are used in general. Since deformation behaviors are coupled in the curved beam, these curved beam elements show the locking phenomena that have a direct effect on the accuracy of solutions. In order to alleviate the locking phenomena in a curved beam element, many studies have been performed by means of various techniques. These are typically the strain element technique [l-3], the selective or reduced integration method [4] and the modified isoparametric elements [5,6], etc. However, these previous works have been confined to beams curved and bent only in the plane of curvature so that no torsion is involved. The original work including out-of-plane deformation, based on independent strain functions, was done by Sabir [7]. However, it has been formulated on the basis of Euler beam theory. Recently, a few works [8-IO] have been done on a general curved beam element, including the effect of shear, which is arbitrarily located in space, showing the in-plane and out-of-plane deformations as well as torsional deformations. These elements have been all formulated on the basis of isoparametric shape function modified in order to alleviate the locking phenomena. This paper proposes a strain element formulation based on the assumptions of the strain fields, which is well known as a means of removing the locking phenomena [I, 7, IO]. By this formulation method, which is based on the Timoshenko’s beam theory, two general curved beam elements are suggested and these elements include the axial, in-plane and out-of-plane shear, bending and torsional deformations. One is a For

analyzing

2. DERIVATlON OF DISPLACEMENT FUNCTIONS IN A LOCAL CURVILINEAR COORDINATE SYSTEM

In a typical curved beam element, the local curvilinear coordinates x, y and z are defined as shown in Fig. 1, and U, v, w and 8,, 0)) 0: are displacement and rotational components in the positive directions of each coordinate axis established at an arbitrary point on the beam axis. The strain-displacement relations on the basis of Timoshenko’s beam theory are obtained as follows: =A!!+!!

“,

dx

R

df?, K,,

=-

dx

d0. KY,=--dx

yl,

=

8, R

2 - 6,,

where R is the radius of curvature of the curved beam axis in the xz plane. In eqn (I), t,, is a tangential 379

Jong-keun Choi

380

Fig. I. Geometry

of a general

curved

and Jang-keun

Fig. 2. Two-noded

beam element

strain, y,, and yYVare shear strains in the xz and zcy planes, respectively. Also, K,,, K,,,. and K,, denote the curvatures in the aforementioned planes. Equation (1) can be expressed in the form of a strain vector:

{z) = [&Ku Yr3&,YvJlT~ where superscript 2.1, Two-noded

T means the transpose

(2) of a matrix.

element

Suppose the strain vector {r} has constant in a discrete curved beam element such as

values

where C,, C,, C,, . C, are arbitrary constants. Eqns (1) and (3) can be integrated to obtain general solutions of element displacement functions as follows:

{4) = PI1 P*ll{C~ = [WC 1.

2.2. Three-noded

{Z} = [C, + C,‘X, c, + c4.x, c, + c,.x,

0 0 0-RqJR

[@,I=

;

“0’ ; _!R ; ;

0

R+

0

0

0

0

0

(0 L

R24 W 0 0 1

0

0

(5)

0 R I

k#d=

0

0

0

0

0

0

0

0

IR 0

c, + c,,..u, c,, + C,z..Y]T.

(IO)

1 0 0

cos+ 0 ; cos.4

sir@ 0 -1 sit-$ R

0 0

0 0

0 0

(7) 1

and

{C } = [C,

c* . C,*]T.

(8) Xl

IYI local

is

(9)

(1) and (9) can be solved to give general of element displacement functions as

(6)

cos+ 0 sin4

-1

Equations solutions foIlows:

c, + c,.x,

0 01_I - sin4 0 cosf$

0

element

Suppose the strain vector {E} has linear variations in a discrete curved beam element such as

(4)

[uv w t3,B, O,]’

curved beam element.

[Q2] represent the strain mode and the rigid body mode in a two-noded general curved beam element, respectively, and {C ) indicates generalized displacements, since it is related to particular point displacements of an element. In eqns (6) and (7) $I denotes xJR. As illustrated in eqn (1) the displacements between in-plane deformation (u, W, 0,) and outof-plane deformation (u, O,, 0:) are decoupled, so that it is easily integrated. Since six freedoms per node exist in a general curved beam element, 12 nodal degrees of freedom are involved in the two noded element in Fig. 2. Thus, we can design a two-noded element easily because eqn (4) has 12 generalized displacements.

In eqn (41,

{q} =

Lim

the element displacement vector in the curvilinear coordinate system. [@,I and

R

Fig. 3. Three-noded

curved

beam element

General

curved

381

beam elements

In eqn (IO).

-R2d

R’(1 -$#J’)

R

0

0

0

0

R

R2

R’4

0

-R'

0

0

0

0

0

-R

-R24

0

fR+b2

0 0

0

0

0 R$

o

0

0

0

0

0

0

0 0

0

R

0

k#J=

R2qb

0

R2 0 R2d

-0

0

0

0

R’C$

0

R2d 0 0

0

0

f R?@

fR'c$'

Rq5

0 R2

0

0

0

0

i (11)

[@41= P21

(12)

{C } = [C, c, . . C,J.

(13)

and

{y} also represents the element displacement vector defined in eqn (5). [Qj] is a strain mode in the three-noded general curved beam element and [@J is the same rigid body mode (zero strain mode) as in the two-noded element, and then {C} indicates generalized displacements in the three-noded element. Since I8 nodal degrees of freedom are involved in the three-noded general curved beam element in Fig. 3, and eqn (IO) has 18 generalized displacements, element can be designed easily due to the consistent form. 3. DERIVATION OF LOCAL STIFFNESS MATRICES

The stiffness matrix can be derived from the total potential energy theorem. The total potential energy of a curved beam element is expressed as

is the Young’s modulus, G is the shear modulus, p2 is the shear correction factor [I I], and A is the cross-sectional area. Thus, b 2GA is the shear rigidity, EJ and EI: are the flexural rigidities in each direction, and GI, is the torsional rigidity. (6) is a strain vector rearranged as it) = [~r.,-Y.~~Y,.KrrlC.,~K.~,lT,

which is related to the stress resultant vector {crL}. It may be also expressed in a matrix form with the generalized displacement vector {C } as

G)=IBlIcJ>

(17)

where [B] means the assumed strain function matrix in eqns (3) and (9). The generalized displacements in eqns (4) and (10) can be related to the nodal displacements by using the element “boundary conditions” in Figs 2 and 3. Thus, its inverse relationship gives the generalized displacement vector {C ) as follows: {C I= [A I-‘&)>

l-P’=;

‘c{~L}T(~}dx -j”jy}T{p,)dr, s \I r,

(16)

(18)

(14) where

where x, and X, denote the initial and final position of a curved element, respectively. The stress-strain relations in a general curved beam element may be expressed as

{AL} = P

[qIq21Tor [q1q2q31T

1= Pl,.2 or PlI.2.3.

(19)

(20)

The subscripts mean the positional substitution of each element nodal points. By substituting eqns (l5), (I 7), (18) and (20) into eqn (14), and taking the first variation of eqn (14) with respect to {AL). an element equilibrium equation in the local curvilinear coordinate system is obtained as follows: [A -‘I’

In eqn (14), JpL} represents a distributed load vector in the local curvilinear coordinate system. In eqn (15) {eL} is called the stress resultant vector in the local curvilinear coordinate system, where mrr m, and rn; represent torsional moment, in-plane bending moment and out-of-plane bending moment respectively. [D] is referred to the rigidity matrix, where E

Q [B]‘[D] s v,

[B]ds [A -‘]{AL} = [A -‘IT

“[@]‘{p,)d.r. s \I

Thus, we can get the element stiffness matrix local curvilinear coordinate system as [KJ = [A -‘I’

“[BITID] [B]d.u [A -‘I. v,

(21) in the

(22)

Jong-keun Choi and Jang-keun

382 4. COORDINATE

TRANSFORMATION MATRIX

OF

AN

Lim

ELEMENT

EQtiATION

Figure 4 shows the description of coordinate systems for a general curved beam element. In order to describe the behavior of a curved beam element in general, it is necessary to define the three sets of coordinate systems in Fig. 4. (x, y, z): local running curvilinear coordinate system (degree of freedom u, I’, II’,0,) 0, , 0,); (x’, y’, z’): local Cartesian coordinate system; (X, Y, Z): global Cartesian coordinate system (degree of freedom U, V. W, O,, 0, , 0:).

Fig. 4. Description

of three coordinate

systems.

transformation matrix [ TJ for a three-noded general curved beam element has an expression of the form

4.1. Tranqformation matrix [T,] between global cartesian set (X, Y, Z) and local cartesian set 6’. y,. z’ 1 As shown in Fig. 4, it is assumed to coincide the x’;’ plane with the .K plane in order to describe an arbitrarily located beam element in the global Cartesian coordinate system. For this assumption, a dummy point k, located on the curved beam element in Fig. 4, is used in producing the local Cartesian coordinate system. The vector ij gives the x’ axis the product vector ik x ij yields the y’ axis and the product vector ij x (ik x ij) yields the z’ axis. In this procedure, a sub-transformation matrix [T,, ] makes the x’;’ plane coincide with the .YZ plane for a general curved beam, which can be transformed from the global Cartesian set (I’, Y, Z) into the local Cartesian set (x’. J“, z’) or vice versa. Thus, the transformation matrix [T,] in a three-noded general curved beam element has the following form:

[T,,l V,,l

Using the [T,] and [Tz] matrices, eqn (19) in the local curvilinear coordinate system can be expressed in the form of the global displacement vector {A(;} of an element:

By means of substituting this expression into eqn (2 I) and multiplying the transpose of [ TJ [T, ] on an element equilibrium the resulting equation, equation is obtained in the global Cartesian coordinate system as follows:

Vi,1

V’, I=

VI,1 where

(23) 4.2. Transformation matrix cartesian set (x’, y’ ,z’) and

[Tz] local

between local cuwilinear .ret

(x3 Y? z) Sub-transformation matrix [Tzl] between a local Cartesian set and local curvilinear set is obtained at an arbitrary point i as

[Tz(=

cos cx, 0 0 I -sin 2, 0

sin c(, 0 1 cos a,

(24)

where x, represents the counterclockwise angle from the .Y’ axis to the .Y axis at the point i in Fig. 4. The

P=l OOlb R=4,953in t =O.O94in b=lin E=10.5x106 v=O.3125 p2 =0.85 WA = 1.244

Ibf/in2

in

General

curved

beam elements

383

1.10 E a” 1.00 P f

0.90

9 % 5 0.60 % 1! 2

0.70

A

Pmthbp(5)

cl

Present ~~cc)

E 5 g 0.60 1

P 0.50

2 Fig. 6. Convergence

curve of the loading

Jdx.

point

deflection elements.

(29)

In these equations, {KG} is the global stiffness matrix and {Fo} is the equivalent nodal force vector of an element.

3

in the case of a pinched

ring with GCSCC

4.3. Evaluation

of a stress resultant vector in the local curvilinear coordinate system

From eqns (1.5), (17) and (18), the stress resultant vector in the local curvilinear set of an element is given hy

{~LI=[DI[BI[A

-‘l&)r

1.00

f 0 ’

0.96

% g ‘J # 0.92 P -0 A B c 0.66 P 0.64 Number Fig. 7. Convergence

curve

of the loading

point

of Elementa

deflection elements.

in the case of a pinched

ring with GLSLC

(30)

.

Jong-keun Choi and Jang-keun Lim

384

where the local displacement vector {AL j is related to the global Cartesian displacement vector {AC) by means of eqn (26). By substituting eqn (26) into eqn (30) the stress resultant vector of an element is obtained in terms of a global Cartesian displacement vector as follows:

P=l R=5 t=1 b=l

Fig.

5.

NUMERICAL

The performance of the two different general curved beam elements suggested in this paper have been studied. Hereafter, the two-noded element will be called a GCSCC (general curved beam element with constant strains and constant curvatures) and the three-noded element will be called a GLSLC element (general curved beam element with linear strains and linear curvatures). Three different computational models will be taken in order to investigate the element performance of the in-plane, out-of-plane and general deformation, including the torsion as shown in Figs 5, 8 and 12. respectively. The first is a pinched ring, the second is a curved cantilever beam and the third is a coil spring. 5.1. Pinched

8. Curved

EXPERIMENTS

ring

Figure 5 shows the geometry of a pinched ring and its material properties. Recent studies [5, 61 confirm that the modification of an isoparametric curved beam element reduces the locking phenomena to a certain extent. In order to investigate the performance of present elements, the same example used in Refs [5,6] has been reworked with the GCSCC and GLSLC elements. Using Castigliano’s second theo-

cantilever transverse

beam under shear load

an

out-of-plane

rem [12], the loading point deflection W’, of the pinched ring shown in Fig. 5 is obtained. The theoretical radial deflection W, is 1.244 in. and the normalized deflection W,(FEM)/W’, (theory) is shown in Figs 6 and 7. Figure 6 shows the convergence trends of the present two-noded element (GCSCC) and the isoparametric linear curved beam element modified by Babu Prathap [5]. It is shown that the deflection calculated by the present GCSCC element converges more rapidly to theoretical values. Figure 7 also shows the convergence trends of the present threenoded element (GLSLC) and the isoparametric quadratic curved beam element modified by Prathap [6]. It is also shown that the present GLSLC element gives the more rapid convergent characteristics and, even at two element divisions, the calculated deflection by this element is almost the same as theoretical values. 5.2. Curred

cuntileoer

beam

under

out-of-plane

trunsz~erse .vhrur load

Figure 8 shows the geometry of a curved cantilever beam and its material properties. For the curved

I

2 1.00

E 3 5

0.95

0 f

0.90

P 0.85 E g

---A

GCSCC

Cl

GLSLC

Theory

0.80

= gi 0.75

Number Fig. 9. Convergence

of Elements

curve of the loading point deflection in the case of a curved an out-of-plane transverse shear load.

cantilever

beam under

General

curved

beam elements

385

1.00

omOXlO-

0.70 0.60 -

E

‘;o.soE

0.40 0.30 A 0.20 0.10

-

0 +

-

0.00 ’ 0.60

I

I

I

’ 4!5.‘w ’

I

90

Angle * Fig. 10. Bending moment curve in the case of a curved cantilever

beam under an out-of-plane

transverse

shear load.

cantilever beam under an out-of-plane transverse shear load P,theoretical solutions are obtained from the Castigliano’s theorem [12]. For the given data in Fig. 8, these give VA = 0.173 x IO-3 rn:=

PR COSI$

m,=PR (1- sin4),

where VA, m, and m, represent the out-of-plane deflection at the loading point, bending and torsional moment in the local curvilinear coordinate system, respectively. In this example, the performances of two elements (GCSCC, GLSLC) have been studied.

-

J-h-Y (4 Div.) GCSCC (8 Div.) GELC (2 Dii.)

A

GCSCC

EI +

0.30

-

0.20

-

0.10

-

0.00

0.00

I

I

I

I 45.00

'

I

l-

90 xl

Angle 9 Fig.

CAS 55 2-B

11.Torsional

moment

(32)

curve in the case of a curved cantilever shear load.

beam under an out-of-plane

transverse

Jong-keun

386

Choi and Jang-keun

Lim 6. CONCLUSIONS

Fig. 12. One coiled helical spring.

Table Element

I. Deflection

at the loading

GCSCC

GLSLC

4

2.87

3.10

8

3.07

3.12

point Theory

II 21

3.13

Simple two-noded (GCSCC) and three-noded (GLSLC) general curved beam elements have been formulated on the basis of the assumptions of strain fields on the Timoshenko’s beam theory. The twonoded element was formulated from constant strain fields and the three-noded one was formulated from linear strain fields. These elements have also been transformed into a global Cartesian coordinate system in order to effectively analyze the general curved beam structures arbitrarily located in space. Numerical results show that the present elements give a better convergence than the modified isoparametric curved beam elements as shown in earlier publications. The GLSLC element gives a more rapid convergence of displacements, as well as stress resultants, than the GCSCC element. REFERENCES

the convergence curve of the normalized deflection at the loading point V,,+(FEM),‘V, (theory). It is shown that the GLSLC element converges very rapidly. Figures 10 and 1I show the normalized bending and torsional moment distributions in the local curvilinear coordinate system. Since the GCSCC element has constant strain fields, the stress resultants do not vary in an element division. Thus, it is necessary to have more element divisions in order to get more accurate stress resultants. Whereas the GLSLC element has linear strain fields in an element division, even two-element divisions give enough accurate stress resultants at any angle 4. Figures IO and I I show that the GCSCC element with eight divisions gives similar stress resultants compared with the GLSLC element with two divisions. Figure

5.3.

9

shows

Spring

A spring becomes the typical example of a curved beam, including out-of-plane general bending, shear and torsional deformations. Figure I2 shows the geometry of a helical spring and its material properties. It shows one turn of the spring with a pitch angle of a = 10’. The theoretical deflection of this spring can be obtained from Castigliano’s second theorem [l2]. This turn is modeled with four and tight element divisions, and the results of the GCSCC and GLSLC element are shown in Table I and compared with the theoretical deflection at the loading point. The GLSLC element gives good numerical results even at four element divisions.

I.

D. G. Ashwell. A. B. Sabir and T. M. Roberts, Further studies in the application of curved finite elements to circular arches. lnr. J. Mrch. Sci. 13, 507 517 (1971). 2. D. G. Ashwell and R. H. Gallagher, Finite Ekwnenr.s for Thin Shells and Curwd Memhrrs. London (1976). 3. J. K. Choi and J. K. Lim, Simple curved shear beam elements. Comm. Nun~er. Me/h. En~ng. 9, 659-669 (1993). 4. H. Stolarski and T. Belytschko, Membrane locking and reduced integration for curved elements. J. Appl. Mech., ASME 49, 172-178 (1981). 5. C. R. Babu and G. Prathap, A linear thick curved beam element. In/. J. Nunler. Meth. Engng 23, 1313 -1328 (1986). 6. G. Prathap and C. R. Babu, An isoparametric quadratic thick curved beam element. In/. J. Nunrer. Met/l. Enp~ 23, 1583-1600 (1986). 7. A. B. Sabir, Stiffness matrices for the general dcformation (out-of-plane and inplane) of curved beam members based on independent strain funclions. In Tile .Mathmurrics o/ Finile Elemrn/s und App/ic~~/wm 2--Proccw/ings (EdIted by J. R. Whiteman). pp. 41 I-421. Academic Press, London (1976). 8. Suman Dasgupta and Dipak Sengupta, Horizontally curved isoparametric beam element with or without elastic foundation including effect of shear deformation. C’ompzct. Slrucr. 29(6), 967~-973 (1988). 9. B. P. Naganardyana and G. Prathap. Consistency aspects of out-of-plane bending, torsion and shear in a quadratic curved beam element. Int. J. Nunwr. .Uc/h. Engng 30, 43 I 443 ( 1990). IO. G. Prathap and B. P. Nagdnarayana. Analysis of locking and stress oscillations in a general curved beam element. In/. J. Nunier. Mrth. E/I~):,~R 30, I77 200 ( 1990). I I. G. R. Cowper. The shear coefficient in Timoshcnko’s beam theory. J. Appl. Mrch. 335 340 (1966). 12. S. H. Crandall. N. C. Dahl and T. J. Lardncr. A!? f,ztroducrion /o (he hlechrmics of Solids. pp. 545-S50. McGraw-Hill, Singapore (1978). 13. T. Y. Yang, Finirc E/cv~~ent S/rwrurul .4mr~w.c. Prentice-Hall. Englewood Cliffs. NJ (19X6).