Simple and Effective Equilibrium Models for Vibration

SIMPLE AND EFFECTIVE EQUILIBRIUM MODELS FOR VIBRATION ANALYSIS OF CURVED RODS

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By A. Benedetti; L. Deseri,:Z and A. Tralli3 ABSTRACT: New finite-element models for curved beam vibration analysis are derived from classical complementary variational principles of elastodynamics. The use of a spline approximation of the axis line (as previously introduced by the writers in the static case) allows for the a priori satisfaction of the dynamic differential equilibrium equations in a simple and effective way. More precisely, starting from the Hellinger-~eissner p~n ciple and making use of a linear interpolation of displacements and momentum fields, a very simple h~bnd mixed model is obtained that can be easily linked with general-purpose finite element packages. Alternatively, fully equilibrated models are derived from the complement~ energy princip~e assuming as unknowns e~ther the momentum or the stress resultant fields; in both cases highly accurate finite element models are obtamed for which upper and lower bounds on eigenvalue estimates are readily available. Several examples are worked out that are capable of showing the efficiency and the wide spectrum of applicability of the proposed method. The comparison with two general-purpose finite element packages of large diffusion let us assess the high level of performance of the complementary energy models for curved elements.

INTRODUCTION The study of the free vibration of curved beams in their initial curvature plane has been a topic of structural dynamics since the last century, when the papers of Hoppe (1871), Lamb (1888), and Love (1944) appeared. In the 1920s, Den Hartog (1928), introduced the Rayleigh-Ritz energy method for finding the natural frequencies of inextensional circular arches with fixed or hinged ends occurring in electrical machinery. Since then a great number of papers have been published providing solutions for noncircular arches, dealing with intermediate supports or taking into account the effect of axial deformation, shear deformation, or rotary inertia. Classical approaches to the analysis of the vibration of rings and arches are satisfactory in themselves, but they do not provide for solutions for arbitrarily shaped curved beams with generic distributions of masses and rigidities and, mainly, they do not allow analysis of complex structures (like a ring-stiffened shell), whereas the finite element (FE) method does. However, the FE analysis of curved structures has always presented great difficulty in static too (Ashwell and Ghallagher 1976; Babu and Prathap 1986); in recent years many researchers have soundly investigated the meaning of the poor convergence rate and the locking phenomena exhibited by the traditional displacement approach based on the principle of m~nimum potential energy. More precisely, the apparently most natural choice of CO (linear) interpolation for the tangential displacements and C l (cubic) interpolation of the normal displacements has led "for deep thin arches to solutions that are not even of the right order of magnitude" (Ashwell and Ghallagher 1976). Obviously these problems get worse in the dynamic case; for instance, when dealing with constant curvature rods, satisfactorily results have been achieved only integrating exactly the displacement fields from the governing differential equiI Assoc. Prof., Inst. of Engrg., Univ. of Ferrara, Via Saragat, 1-44100 Ferrara, 1taly. 2Res., Inst. of Engrg., Univ. of Ferrara, Via Saragat, 1-44100 Ferrara, Italy. 'Prof., Inst. of Engrg., Univ. of Ferrara, Via Saragat, 1-44100 Ferrara, Italy; formerly, Dept. of Constr., Univ. of Firenze, Piazza Brunelleschi 6, 50121 Firenze, Italy. Note. Associate Editor: P. K. Banerjee. Discussion open until September I, 1996. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on October 3, 1994. This paper is part of the Journal of Engineering Mechanics, Vol. 122, No.4, April, 1996. ©ASCE, ISSN 0733-9399/96/0004-0291-0299/ $4.00 + $.50 per page. Paper No. 9362.

librium equations (Davis et aI. 1972). Recently, a more general but rather complex element, able to solve static and dynamic three-dimensional rod problems was presented (Tabarrok et aI. 1988). Namely, by evaluating the strain displacement relations for zero or constant strain distributions, the rigid-body and constant-strain displacement modes of the rod are obtained; these transcendent displacement functions are then used as a sound basis for interpolation. As is well known, selective and reduced integration techniques (Babu and Prathap 1986), multifield variational formulations (Saleeb and Chang 1987; Reddy and Volpi 1992) or curvature-based elements (Lee and Sin 1994), proved to be capable of overcoming the quoted difficulties in the static elastic case; indeed, the evaluation of the performance of these approaches in the dynamic case has received until now little attention. It is worth remembering that in the early days of the FE method, a number of planar curved equilibrium elements were developed (Tezcan and Ovunc 1965; Morris 1968; Ping 1969; Argyris and Sharpf 1969). Unfortunately, these formulations, successfully and consistent for straight beams and frames (Elias 1972; Ghallagher 1975), are all confined to a particular arch geometry and worked out from the flexibility approach, and the stiffness matrix can be worked out anyhow from inversion. Moreover, because in this treatment displacement shape functions play no role, it was impossible to obtain consistent load vectors and mass matrices, forcing one to resort to static load and inertia force lumped representations. In recent papers by the writers (Benedetti and Tralli 1989; Alessandri et aI. 1989), the first limitation has been overcome through the substitution of the axis line with a B-spline approximated one; according to the classical works of T. H. H. Pian, a hybrid equilibrium model was developed, arriving so directly to the computation of the "exact" stiffness matrix (apart from the numerical integration errors), and consistent (in a variational sense) load vectors. The aim of the present paper is to show that equilibrium models derived from the classical principles of the elastodynamics (Toupin 1952; Washizu 1966, 1982; Tabarrok 1984), yield great accuracy in the vibration analysis of arbitrarily curved beams, in a simple and efficient way. With this in mind, the basic kinematical and mechanical relations for linear dynamics are reviewed in the following section with relation to a general model of a shear deformable thick beam with large variable curvature. Moreover the spline approximation of the axis line allows one to directly obtain in an explicit integral form stress resultant fields satisfying equilibrium equations. Next, hybrid models derived on the basis of the HeIIingerJOURNAL OF ENGINEERING MECHANICS / APRIL 1996/291

J. Eng. Mech. 1996.122:291-299.


Reissner principle (Washizu 1982) are introduced. Nevertheless, while the hybrid stress-assumed models lead to a nonlinear eigenvalue problem it is noteworthy that, for the problem at hand, the assumption of CO continuous displacement and momentum fields, together with stress resultant fields satisfying a priori indefinite static equilibrium equations, attains the solution through a standard linear eigenvalue problem. Then equilibrium models derived from the complementary energy principle, or equivalently by the complementary Rayleigh quotient (Tabarrok 1984), are stated. Numerical tests show that truly satisfactory results can be achieved with both the hybrid and the equilibrium models for thin or thick arches of any shape, assuming either a constant or linear interpolation for momentum components. As a matter of relevance, in the case of equilibrium elements, good upper-bound estimates for eigenvalues come out from a standard eigenvalue problem.

BASIC RELATIONS Kinematical Assumptions A deep curved beam model allowing for the effects of transverse shear is employed. In this model the classical hypothesis that plane sections remain plane after deformation (extended by Winkler in 1858 to deep arches), is retained; however, the sections do not necessarily retain their normality to the axis line in the deformation process. This model represents a particularization to the monodimensional case of the higher order shell theory developed by Naghdi and Reissner; nevertheless, in the present model radial stress and strain are neglected. Let us consider a generic curved element (Fig. 1), referred to a Cartesian coordinate system O(x, y, z); the z = 0 plane contains the centroidal axis and the external loads, and is a symmetry plane for the rod. Further, a curvilinear abscissa s(O, l) spans the axis line, which is known either by a parametric representation x(s), y(s), or by an explicit one y = y(x). A local reference system O'(~(S), T1(S)) is defined according to the tangent and the normal of the axis at each abscissa, and the initial curvature radius, supposedly variable with s, is denoted by R(s); the curvature K of the axis line is then defined as the reciprocal of the local curvature radius. The vectors u

T

= [u,

v, e];

b = [b~, bTl' b..J T

(1)

represent the displacement and the distributed load (per unit length) vectors; they are referred to the local reference frame. The normal and shear strain components e, 'Y, in a material fiber at a distance TI from the reference surface, are given as £

= (Eo -

1

T1X) - - ; I _.:!l

R

where Eo and 'Yo

'Y

=-'Yo1

(2)

_.:!l

x = the change in curvature. Therefore, when the displacements are assumed to be small, the strain displacement relations are Eo =

U,I -

'Yo =

KW;

KU

+

w... -

e;

X=

e.,

(3)

where ( )... and () denote derivatives with respect to curvilinear abscissa s and time t, respectively. Denoting by E T = [Eo, 'Yo, X] the generalized strain vector, the previous relations can be written in a compact form E

(4)

=Tu

where T = the compatibility operator, so defined. Furthermore

m = pA;

me = pJ,

(5)

define the inertia terms associated to the kinematical model assumed, depending on the density p, the area A, and the inertia moment J, of the cross section, respectively. Assuming the normal and shear forces N and V, and the bending moment M as generalized stress resultants aT = [N, V, M]

(6)

the equations of motion can be written in the form T*a

+ mo = b

(7a)

where T* (the adjoint of T) = the equilibrium operator; and m = diag[m, m, me]. In dealing with complementary variational principles in elastodynamics the momentum field pT =

m'o =

£p~, PTI' p..J

(7b)

is usually introduced. Therefore the motion equations can be written in the equivalent form T*a +

p= b

(7c)

Finally, for a homogeneous isotropic Hookean material by performing the integration onto the section area of (2), the elastic compliance matrix n- I can be easily computed, yielding so the constitutive relation

(8) Equilibrated Stress Field and Spline Approximation Often in engineering practice we are dealing with arches or other curved beams whose axis line analytical form is not given; in this case it is always possible to define the structure through the coordinates of a certain set of points. In the general case, a cubic B-spline approximation )i(x) of the axis line is used

R

= strains at the beam reference surface; and

u

5

=

.. >0

4

y

3

2

o Knots

x FIG. 1. Reference Systems 0 and 0' for Arbitrary Beam Element

1-3 1-2 1-1 i 1+1 1+2 1+3 FIG. 2. Ba818 of Cubic B-apllne Interpolation [The Value and Derlvatlve8 of an Approximated Function at a Knot (I) Depend Only on Weighting Functions B~x) with (/- 2) < k< (I + 2)]

292/ JOURNAL OF ENGINEERING MECHANICS / APRIL 1996

J. Eng. Mech. 1996.122:291-299.

.+1

y(X) =

2: XiBj(X)

(9)

Tabarrok 1984), which can be written, for the time-dependent problem at hand, in the form

;'-1

L{L (-~pT.m-l.p +


12

where X-I> .•• , X.+l are the coordinates of the spline nodes (or more precisely "knots"), and B_1> ... , B.+ 1 are the basis functions defining the spline (Fig. 2). The most noticeable properties of spline approximations in this context, apart from the sound theoretical basis, appear to be that (I) the numerical implementation is really easy and effective; and (2) the values of the derivatives are explicitly given at the knots. It is worth remarking that in a case of geometries such as circles, ellipses and hyperbolas, the use of rational B-spline (instead of the standard ones) could allow exact modeling of the curve. On the basis of elementary static, a stress field, which exactly satisfies the dynamic equilibrium equations, (7), can be immediately stated. More precisely, the stress resultants induced at a generic curvilinear abscissa s by any given external load referred to the element coordinate system O(x, y, z) and applied at S, can be written making use of a force transfer matrix. By setting the external forces at instant t as (10)

I)(t/ = [FAs, t), Fy(s, t), M,(s, t)]

we obtain

O'°(s, t) =

N(S,t)} V(s, t) = { M(s, t)

I

Y.x

g

g

I

_Y.x g y(s) - y(S)

o

-x(s)

+

(11)

where g (the gradient of the axis line) holds Vi + Y~x' Therefore the stress field induced by any distribution of applied loads b(s, t) or inertia forces -m(s)' ii(s, t), can be easily expressed as an integral form of the matrix L(s, S)

L L

=

O'/(s, t) = -

LT(s; S). b(s, t)· L(s, s) ds

-

O'T.

pT. g

TU) ds + O'i' (UI -

OI)lr•.} dt

= Ab

LT(s, S)'p(s, t)·L(s, s) ds=

-Ap

(12a)

(12b)

In the stated relationship the matrix LT(s, S) accounts for the transformation of applied loads from the local to the global reference frame system, and A denotes the transfer operator of the applied loads [field transfer matrix, Fujii and Gong (1988)] for the spline-approximated curved arch. The complete representation of the stress field is then built up superposing the particular solutions, (12), to the homogeneous one (1 I). Obviously the resultants obtained in this way satisfy, for the given arch, the equilibrium equations; then the convergence to the exact equilibrium is asymptotic whenever the approximating curve converges to the actual one (i.e., when the number of the spline knots becomes sufficiently large). Therefore, in computing the stress resultants at the spline nodes, the advantage of using the spline approximation of the axis lines is apparent. In particular it is possible to model curved beams of any shape (whether analytically known or not) with exactly the same algorithm (Benedetti and Tralli 1989). HYBRID-MIXED MODEL

As is well known, multifield FE models can be derived from the three fields Hellinger-Reissner functional (Washizu 1982;

(13)

where the prescribed external forces b are neglected, a/ denotes the prescribed displacement components (a, lV, 6) and r ul lists the element end sections where the relevant kinematical boundary conditions hold. Integrating by parts with respect to the time and space derivatives of the displacement components, we obtain the following equivalent functional:

L{L 12

'lTMO', u, p) =

- uT·(T*O' +

e

P)]

[

-~ pT. m-

I

•

p +

ds - O'j'Ollr..} dt

~ O'T. n-

+

L

1

• 0'

pT· U dsl::

(14)

the last term in (14) will have no influence in variations of 'lT~ if we assume that both U and p are prescribed at times t l and t2 • Looking for a free-vibration analysis, a time-independent functional can be derived assuming a harmonic time dependence of the three fields; for instance in the form Po(s)' cos wt,

O'(s, t) (15)

= O'o(s)' sin wt x(S)

{I)} = L(s, S)1)(t)

O'b(S, t)

+ ~ O'T. n- I . 0'

u(s, t) = uo(s)' sin wt; p(s, t) =

o

g

'lTR(O"U,P)=

where w denotes the circular frequency of vibration. Then, substituting in 'lTk and specializing to a generic element of length l' with prescribed displacements iio at the ends, we arrive at the following stationary functional 'lT~. However, the instants t l and t 2 are arbitrary; so they can be spaced without loss of generality by an interval of length 2'lT/w that cancels all trigonometric function integrals. The cited functional is

L

'lT~(O'~, u~, p~) = (-~ p~T. m- p~ + w· p~T.u~ I

+ ~ O'~T·n-l.O'~ -

•

U~T'T*O'~) ds - O'~T·Uolr..

(16)

Following Benedetti and Tralli (1989), a stress field satisfying the homogeneous equilibrium equations is established according to (II) element-wise, in terms of element stress parameters pe. Assumed stress hybrid models (Washizu 1966; Tabarrok 1971) require stress fields that a priori satisfy the dynamic equilibrium equations (7b) such that O'~(s)

= L(s, s)· W -

wAp~

(17a)

where p~ = the momentum vector interpolating field; however, substituting in (16) a nonlinear eigenvalue problem is achieved. Hybrid-mixed models can be constructed (Pian et al. 1983) under the weaker assumption that the stress interpolating fields must satisfy only the homogeneous part of (7) O'~(s)

= L(s, s)· W

(17b)

The displacement vector assumed CO continuous, is interpolated in terms of the generalized nodal displacements qeT

= [Uh Wh 910 U2' W2'

92];

U

= ~'qe

(18a)

JOURNAL OF ENGINEERING MECHANICS / APRIL 1996/293

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where matrix cf» collects the standard Lagrange linear shape functions. For the sake of simplicity (but a different choice holds, too), the same discretization can be assumed for the momentum field, writing down r,T=[pt,p 2]; P=Cf»·P· (l8b) Therefore inserting (17), (18), and (19) into the functional the following function is obtained:


'TT~

(19)

where

are symmetric positive definite matrices and Hand G are the same as defined in Benedetti and Tralli (1989). The stationary conditions with respect to r', W, and q' yield -A

[

o w·e

w.e]

0 H T

-G'

-G

T

0

[r'] W

(21a)

=0

q'

otherwise. eliminating the stress and momentum variables we run to the standard relationship -K"q'

+ w2 ·M'·q' = 0

(2Ib)

where K' = GT·H-1·G; and M' = eT·A-1·e The following comments are due: • The assumption of the same functions for the displacement and momentum fields implies that the relationship P = m' Ii turns out to be pointwise satisfied; therefore the presented statement could be derived in a shorter but less general way a priori imposing this condition. • It is easy to verify that in the case at hand matrix M' coincides with the usual consistent mass matrix of a [!J model. whereas the stiffness matrix is the "exact" one (Benedetti and Tralli 1989). SINGLE-FIELD EQUILIBRIUM MODELS

The principle of stationary complementary energy can be immediately obtained by the Hellinger-Reissner functional 'TTk, (14), introducing the hypothesis that generalized stress (J' a priori satisfy the motion equation (7b) (Washizu 1982; Tabarrok 1984). In particular, if the dependence of the stress components a(s, t) versus pes. t) is assumed explicitly as stated in (11) and (12b), assuming for the momentum field description any simple harmonic function of time, and noting that free vibrations do not depend on nonhomogeneous kinematical boundary conditions (k.b.c.), the following single-field time-independent functional is derived:

L

'TT~(Po) = (-~ p~·m-l·po + ~,w2.a~'D-I.ao) ds

Momentum models are based on a FE discretization of Po (and consequently of Qd, which can be carried out in a standard way. The stationary conditions of 'TTb represent the compatibility equation between the displacement associated with the momentum and the deflection induced by the inertia forces; in this sense (Washizu 1982). the complementary energy approach appears to be quite equivalent to the classical force method in structural dynamics. It is worth noting that. in principle. any piecewise constant basis function can be employed, taking care that Po components be square integrable functions. However, as suggested by Grammel (1939), whenever the assumed coordinate functions a priori satisfy the kinematic boundary conditions (KBC), the computed eigenvalues, which obviously are upper bounds of the squares of the exact circular frequencies, are smaller than those based on weaker assumptions, and hence closer to the solution. Therefore, in developing numerical examples. the standard linear Lagrangean basis functions as in (18) were used, allowing so for a priori satisfaction of the KBC. As a consequence. the kinetic energy term appearing in the numerator of Qc can be constructed summing up the generic element contributions given by 1/2 r'T. A . r' with A defined by (20). The advantage of introducing the spline approximation of the axis line together with a Gauss numerical integration scheme becomes apparent in evaluating the matrix in the denominator of Qc; namely the proposed method can easily deal with arbitrarily curved arches either thick or thin and with variable section too. Eqs. (11) and (12b) give the stress field for a given arch in terms of the momentum vector Po and an arbitrary set of hyperstatic reactions. It is worth noting that because redundant reactions do not occur in the kinetic energy term [(24), the numerator of Qcl, they do involve only zero frequency modes (Tabarrok 1984); it is therefore always possible to carry out a static condensation of these unknowns. For sake of completeness it is to note that the denominator of Qc can be evaluated also in a quite simpler but more timeconsuming way, directly solving for the stresses at the Gauss points the whole structure loaded in turn by each of the discretized inertia forces, and after performing numerically the stress integrals. Equilibrium elements in terms of stress, developed by de Veubecke (1965) for static problems, can be derived by the complementary energy functional, (23), and are suitable for free-vibration analysis. The associated model can be worked out making use of the expression of the momentum fields in terms of internal stresses and assuming for instance a constant distribution for Po in each element. In fact, rigid-body equilibrium equations for a curved

-11----- dX ----r-

y

t

(22)

The eigenvalues 00 2 can be conveniently expressed in terms of the so-called complementary Rayleigh quotient (Tabarrok 1984) ('

2

QC() C

Gl

Clamped Quarter-Circle Arch

~

This common practice example (Fig. 8) is worked out varying once at a time both the element subdivision and the slenderness ratio [for a deep discussion see Wolf (1971)]; by this way it is possible not only to show the relative convergence ratios of the various models, but also to assess the ability of the proposed elements to deal with thin or thick structures with the same accuracy. In Fig. 9 the rate of convergence of (constant momentum) equilibrium element, hybrid-mixed, and ABAQUS elements are compared for the case of slenderness ratio of 50; also in the case of a thick arch, linear momentum and ABAQUS elements show a high convergence rate. A previous numerical solution taking into account the rotary inertia but neglecting the shear deformation is available in Wolf (1971). Moreover Wolf presented a plot illustrating the

. . .

ICT Gl IL

.!

~ ()

240

160

(3

.E'tl

80

Cl:

~

e

0 0

20

40

60

80

100

Slenderness Ratio R/I FIG. 10. Frequency Crossover Plot as Function of Slenderness of Arch for Example 3


J. Eng. Mech. 1996.122:291-299.

o

II)

r...:

E=1

1

b=1 P= 1

h

variable

b(l'l)


FIG. 11.

Input Data for Example 4: Thick Circular Ring

J~~o 4.0

FIG. 13. Input Data for Example 5: Hinged Parabolic Arch with Variable Cross Section

Free Vibration of a Thick Ring Shear Def. & Rot. Inertia Inc!. 0.8

WjR2{"f

>-

C

•c:r • ~

i

~

~

~

0,4

U

=

/

0

~

'I:l c(

10 ...- - - - - - - - - - - - - - - - - - - - . ,

D 0.8

IL

.!

First Natural Mode Shape

D 0

()

0,2

0,0

V

0,0

V

Y 0 C

• 6

• 0,5

-

E .....

~

c:::

o

Kuhrs Experiment Roo & Sundararajan Seidal & Erdelyl SAP V • 200 Quadr. EI.

8

6

CII

> .!!

4

w

-

o

2

Hybrid Mixed. 4 Elmt.s

6

Compl. Energy. 4 elmt.s

•

a l-.-,,....,.....~:;::;:::;:;::::;:;;:::;:::;:;::;::;::;:;::;:::;::;::;~..........T""'""""~~

I 1,0

Arch Axis Une ABAOUS 64 Three Node B.s Hybrid . Mixed 16 EI.s Unear Momentum 16 El.s

1,5

o

2,0

7

Slenderness Ratio h/R

21

14

Abscissa

FIG. 12. Natural Frequency of Thick Ring as Function of Depth versus Radius Ratio

28

35

42

(m)

Third Natural Mode Shape

10

-r--------:--------------,

Thick Circular Ring The free-vibration analysis of a thick circular ring (Fig. II) is a deeply discussed problem in the technical literature (Hoppe 1871; Love 1944); moreover, by virtue of its large use in machinery parts, experimental results are available too (Kuhl 1942). Fig. 12 shows the variation of the first natural frequency as a function of the depth versus radius ratio, as evaluated by means of the two proposed models or using a standard FE package for beam or plane stress analysis, in comparison of existing analytical and numerical results (Rao and Sundarajan 1969; Seidel and Erdelyi 1964), More precisely the analyses were performed using a quarter circle discretization of either four beam elements, or 200 plane stress quadrilaterals. As is apparent, the prediction of the complementary model is always nearer to the experimental solution than that of the other beam models; however, it is worth mentioning the discrepancy highlighted by the experimental result with respect to the plane stress analysis, too, in the field of very thick rings.

Hinged Parabolic Arch Fig, 13 illustrates a parabolic hinged arch whose cross-section width varies according to the inverse of the cosine of the meridian angle. It has the following characteristics: (1) elastic properties: E = 10,000 MPa, v = 0.0; (2) mass density: m = 2,500 kg/m 3 ; (3) axis line: y = (42x - ~)/58.6 m; and (4) section dimensions: h = 1.0 m, b = 4/(cos {}) m. The lower four eigenvalues of the arch obtained either by means of each of the three models previously discussed dis-

E ..... c:::

o

8

6

III

~

4

W 2

7

14

21

Abscissa

28

35

42

(m)

FIG. 14. Hinged Parabolic Arch as Function of Mesh Subdivision (a) First Natural Mode; (b) Third Natural Mode

cretizing the arch line into 16 elements (44 degrees of freedom) or with a solution carried out using ABAQUS program with a mesh of 64 B22 elements (380 degrees of freedom) are in full agreement. In Fig. 14 the first and third vibration modes are compared; the eigenvectors computed through the proposed models appear to be very close to the ABAQUS solution, denoting also high precision in locating the position of the free-vibration fixed points.

Logarithmic Spiral Suspension Rod Fig. 15 illustrates a highly idealized suspension rod whose axis line is expressed as a logarithmic spiral; the section width JOURNAL OF ENGINEERING MECHANICS / APRIL 1996/297

J. Eng. Mech. 1996.122:291-299.

100

els based on multifield variational principles, demonstrating by far its superior versatility.

50

IV/ f

-50

~ P'''

\

-100

\l\\~

-150

~~ ~

-200


CONCLUSIONS

I~

o

• .-.

"'"'"'"--- t-I~ ~ t-.....

-250 -300

-50

o

50

.........

"

-------r---

100

150

200

250

300

FIG. 15. Geometrical Data for Example 6: Axis Line and Width Variation for Logarithmic Spiral Suspension Rod TABLE 2. No. of elements x2 (1 )

Variable Curvature Suspension Rod

SAP Hybrid mixed (2)

EqUilibrium model (3)

ABAOUS ABAOUS

2-node beam" (4)

823 8eamb (5)

822 8eame (6)

(a) First natural frequency

2 4 8 16 32

211.84 201.84 192.78 190.95

-

229.44 201.85 192.59 190.86

-

-

-

-

199.87 188.36 185.65

190.84 190.36

247.10 195.83 187.09 185.32 184.95

(b) Second natural frequency

2 4 8 16 32

334.26 366.08 370.56 374.84

-

453.11 409.47 379.21 373.60

-

373.27 371.06

356.48 344.21 343.26

385.27 318.62 287.58 282.45 281.60

(c) Third natural frequency

2 4 8 16 32

111.535 967.95 759.Q7 758.56 -

745.36 780.01 802.90 780.19

-

-

765.70 765.11

716.70 686.15 686.09

828.03 560.22 522.17 520.10 520.05

·Standard BemouIli-Navier thin straight beam with linear axial and cubic transverse displacement fields. 'Standard Bernoulli-Navier thin straight beam with linear axial and cubic transverse displacement fields. 'Curved parabolic thin or thick three-node beam with constant section.

varies according to the square root of the curvature radius and = 1.5eq> mm, a = 3 (0 p)ll2 mm, b = 100 mm, E = 210,000 MPa, and m = 7,850 kg/m3 • In Table 2, the results of the computed eigenvalues are compared; due to the simultaneous variation of the curvature and the rigidity, only the proposed models are capable of catching the result with less than 16 elements for the whole structure. Moreover, while the constant section of ABAQUS beams can help in understanding the rough behavior of poor meshes, it appears at least questionable that for very fine meshes ABAQUS results disagree not only with other programs but also between themselves. However, following the line of deduction of the global equilibrium equations, (25), we realize that in this particular case also the constant momentum assumption is a poor approximation. Indeed, in this case the spline-interpolated hybridmixed model highlights the robustness of finite element modp

In this paper, a set of stress-based variational principles for elastodynamics was discussed; for the special case of problems that allow for a partial or total satisfaction of the indefinite dynamic equilibrium equations, we arrive respectively at hybrid-mixed or complementary energy FE formulations. Making use of a spline-interpolated geometric field previously introduced by the writers, a family of curved two-dimensional beam elements is derived, featuring the a priori solution of the differential equilibrium equations. Then, starting from the Hellinger-Reissner two-field principle and requiring only the homogeneous solution of the equilibrium vector field, we set a hybrid-mixed element naturally locking free that can be linked very easily to existing FE programs. It allows for thick to thin sections and, in the case of curved beams, outperforms conventional straight and curved two-node elements providing good approximate solutions with only a few degrees of freedom. Making use of example 6, we showed also that, in case of very complex geometry variations, only elements allowing for both curvature and section variation, are able to predict sensible values of the eigenvalues with a very poor mesh. Unfortunately, this is not allowed in the used version of ABAQUS package, despite its being a major reference computational tool. Alternatively, introducing a complete solution of the equilibrium vector field in terms of either stress resultants or momentum components, we run to flexibility approaches based on the complementary energy principle. In both cases we obtained FE models of very high accuracy; their extensive test throughout the presented examples outlined some advantages and some difficulties that are discussed later on. The most remarkable quality is that equilibrated FEs show a uniform convergence to the exact solution with a very high rate. As a consequence, making use of well-known bounding formulas, reliable limit intervals can be drawn. Of the number of drawbacks it is noteworthy that the force formulation of these methods withstands the inclusion of the developed elements in standard FE codes and does not allow for a generalized extension to the solution of two- and threedimensional elasticity problems. However, the worked examples pointed out the robustness of curved beam elements so far derived from the complementary energy principle; in fact, they showed convergence rates similar to the three-node curved elements, although these last involve nearly twice as many degrees of freedom as the proposed elements. Finally, the presented numerical tests encompassed several standard practice problems that are common in mechanical and civil engineering, demonstrating the wide applicability of equilibrium-based FE models.

APPENDIX.

REFERENCES

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APPENDIX II.

NOTATION

The following symbols are used in this paper:

A, C = positive definite symmetrical matrices; B= B-spline basis functions; b~, bTl' bmz = vector of generalized distributed loads; D= constitutive matrix of the material; flexibility matrix; F f= vector of end-generalized nodal forces; g = axis line gradient; K= stiffness matrix; L= force transfer matrix; I = arc length; M= bending moment; M= mass matrix; m, me = distributed translational and rotational mass; N= axial force; O'(~, "1) = local corotated reference frame; O(x, y, z) = global reference frame; p = momentum vector field; Q= Rayleigh quotient; R = initial curvature radius of axis line; r = vector of generalized nodal momenta; curvilinear abscissa; s time variable; t T, T* = compatibility and equilibrium differential operators; stress parameter Boolean transformation matrix; vector of generalized displacement fields; u, v,-6' shear force; y(x) = approximated axis line; z= global (element) equilibrium matrix; p = generalized stress parameters; £, 'Y = axial and shear deformations; K = initial curvature of element axis line; A= integral field transfer matrix; A = eigenvalues of dynamic discretized problem; fl= reciprocal mass matrix; 1T= energy functional; p = mass density; cr, T = normal and tangential stresses; cl> = shape functions of displacement interpolations; X = change in curvature of axis line; and w circular frequency.

= = u= = v=

Subscripts and Superscripts C = complementary; e = element; mz = relative to axis perpendicular to plane; o = referred to neutral axis line; R = Hellinger-Reissner; T = transpose of a matrix; and ~, "1 = referred to corotated versors.

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J. Eng. Mech. 1996.122:291-299.