On a hierarchy of conforming timoshenko beam elements

CompuIcrs & S~rucens. Printed in Great Britain.

w45-7949/81/030335-loso2.cw0 @ 1981 Pergamon Press Ltd.

Vol. II. No. 3-4. pp. 335-344.1981

ONAHIERARCHYOFCONFORMING TIMOSHENKOBEAMELEMENTS A. TESSLERt Structural Mechanics Research Department, Northrop Corporation, Hawthorne, California, U.S.A. and S. B. DON& Mechanics and Structures Department, University of California, Los Angeles, California, U.S.A. (Received 23 Sepfember 1980; received for publication

18 December 1980)

Abstract-Presented herein is a hierarchy of beam elements which include the effects of transverse shear deformation and rotary inertia. Interdependent interpolations are employed for the deflection and bending rotation fields to obtain a family of virgin elements. By imposing a continuous shear constraint condition on each member of the virgin element family, a series of constrained elements can be generated. Stiffness and consistent mass matrices and consistent load vectors are formulated by Gaussian quadrature using formulas for exact integration. These elements exhibit excellent modeling capabilities and suffer no deficiencies in the range of small thickness/length ratios. One important feature of certain constrained elements is that it is possible 10use them to explain fhe essence of the “reduced integration” stiffness matrices that have appeared in the literature. INTRODUCTION

A number of beam-type finite elements based on Timoshenko theory have appeared in the literature. Such elements extend the modeling capabilities beyond those of classical beam theory because of their inclusion of rotary inertia and transverse shear deformation. One such element due to Hughes et al.[l] recently, which relies on selective reduced integration, obviates “locking” (i.e. excessive shear stiffness) and permits applications far into the thin beam regime without numerical difficulties. Although the manifestations of selective reduced integration are clear and desirable, the procedure in many respects appears magical. An explanation is given herein by means of a comparison of this element with one from a hierarchy of conforming Timoshenko beam elements using what will be called interdependent variable interpolations. In addition, consistently derived mass matrices and load vectors, which were not addressed by Hughes et al. are presented. For reference in the sequel and for establishing notation, the equations of Timoshenko beam theory [2,3] are first summarized. Consider a x-y Cartesian coordinate system with the x-axis coincident with the beam’s neutral axis. The two generalized displacements employed are the transverse displacement w and the bending rotation 8. The difference between w.~ and 0 is the definition for the generalized shear angle y, i.e. y=w,-0.

(1)

The corresponding generalized forces are the moment M and shear force Q. The stress-displacement relations and equations of motion are, respectively: M = -E&I.,;

Q = k*GA(w., - 0) = k*GAy

(2)

[k*GA(w., - 0)l.x + q = pA+ [EIB,],~ t k*GA(w., - 0) t m = ,oI(i tSenior Engineer. OProfessor of Engineering and Applied Science.

(3)

where q, m-distributed transverse force and moment loadings; E, G-extensional and shear moduli; p-mass density; A, I-cross-sectional area and moment of inertia; k*-shear correction factor. The boundary conditions at an edge of a beam are: w = \t

(1)

or

k’GW(w., - 0) = 0 (4)

(2) either

8 = B or -E18., =fi

where a superior bar denotes a prescribed quantity. Hamilton’s principle embodying this theory can be written as: {pAk2 t pld} dx

6

-;bL t

I

(El0: + k*GA(w, - Q2}dx

oL{wqtOm)dn]df=O.

The first Timoshenko-type element was due to McCalley[4], who developed a two-node four dof (degrees of freedom) element. This was extended to tapered beams by Archer[S]. Severn[6] and Davis et al.[7], using slightly different approaches, arrived at the same or equivalent results as McCalley. The commonality among these formulations is that they begin with a displacement characterization as the sum of a bending deflection wb and shear deflection ws. By means of a statical moment-shear equilibrium condition, these two deflections can be combined into one. In this connection, mention is made of a similar procedure used by Egle[8] to arrive at an approximate version of Timoshenko beam theory that was employed by others in their beam element formulations. The element’s nodal dof are the total deflection w and the bending rotation 0, the latter defined as dwb/dx. Many terms in this stiffness matrix contain the factor 4 = 12EI/k2GA12, where 1 is the element’s length. A convenient feature of this 335

A. TESSLER and S. B. DONG

336

stiffness matrix is its straight-forward reduction to that for classical beam theory. When the shear rigidity is large in comparison to the bending rigidity, then 4 +O and the stiffness coefficients immediately assume the classical beam forms. A detailed derivation of this element may be found, for example, in Przemieniecki[9]. Note that this finite element formulation does not utilize Hamilton’s principle in the form given by eqn (5). Beam elements based on variational principle (5) include those by Carnegie et al.[lO] and Dong and Wolf [ 1I]. A highly attractive feature with this approach is that only COcontinuity of w and 0 is required for full interelement continuity. Carnegie et al. used cubic interpolations for both w and 0 to produce a four-node, eight dof element, while Dong and Wolf employed quadratic interpolations for both variables for a similar type element except with three nodes. Linear interpolations lead to an element exhibiting overly stiff characteristics, which gives reasonable results only for the case of very thick beams. It was not until Hughes et a/.[l], using selective reduced integration, that a two-node element with linear interpolations evinced satisfactory properties over a wide range of thickness/length ratios.

The remaining elements may be grouped together as “higher order” elements. They involve interelement continuity of more nodal dof than those dictated by variational principle (5), for examples: w,,, fJ.,,y, etc. Those belonging to this group include Kapur[ 121,Nickel1 and Secor[l3], D. L. Thomas et al.[l4] and J. Thomas and Abbas[lS]. Their elemental attributes are summarized in Table 1. Aside from their quest for better modeling capabilities, some of these investigators anticipated greater latitude in prescribing boundary conditions with the additional nodal dof. However, these attempts go beyond that which is called for by this theory. However, some of these formulations are not proper as they impose higher continuity than is appropriate, such as at a discontinuity in cross-sectional properties. Additional comments of this nature may be found in Refs. [IC161. Herein, a hierarchy of elements is set forth on the basis of interdependent variable interpolations. Then from these elements, called virgin elements, a number of constrained elements can be obtained by imposing various orders of continuous shear constraint. Transformations predicated on such constraints allow for the

Table 1. Elementconfigurationsand interpolationorders Kinematic

of Nodes, dof)

(No.

Variables Element

[Refl

Interpolation

Nodal Configuration

Order or Equivalent

(2. 4) 0

w - linear Hughea et al [ti McCalley Archex

141 [s]

Severn Davis et al

(2, 4)

wb - cubic W

1;

a

-

(4, 81 0

e - cubic Dong and Wolf

w - quad. 0 - q;lad.

(3. 6)

Ill1

- cubic

I121

;

(2, 8)

Kapur

-

0

0

linear

Carnegie et al[lol w _ cubic

Nickel1 and secor I131

a

0 - linear

=

z

4

0

0

\

1

cubic

w - Cubic 0 - quad.

(3, 7)

w - cubic

(2, 8)

l

l

"

TIM4 D. L. Thanas et al

1141

J. Thoma8 and Abbas

1151

l

Y - linear

v

(2, 8)

w - cubic

0

0

8 - cubic

Legend for Nodal Degrees of Freedom

l -

WI Q

n = WI W,.&lQ

AIWb'

wb'x' wa; we;*

XI

e

v=

0

Wlxl

wte;

Y

-

WI

81

er,

337

On a hierarchyof conformingTimoshenkobeam elements direct formulations of stiffness and consistent mass matrices and consistent load vectors of the same order. Numerical examples for static and vibration analyses are given to confirm the effectiveness of these elements. Then it is shown that the stiffness matrices based on selective reduced integration are identical to those of a particular subset of constrained elements. An explanation for this is given. INTERDEPENDENT VARIABLE INTERPOLATIONS

where

To understand the basis of what is called interdependent variable interpolations for Timoshenko beam element formulations, consider the shear angle y as given by eqn (1). For negligibly small shear deformation, angle y should vanish. This may be regarded as a Kirchhoff constraint, i.e., w.x = 0; xe(O,I).

(6)

Suppose that the same order interpolations for w and 0 are used, i.e., P+l

w=

C

P+’

a&“;

n=cl

e= nzo bnt”

(5 = xll).

(7)

Upon enforcing Kirchhoff constraint (6), there results: bo-a,;

b,=2as;

. . . . . . b,=(p+l)a,+l.

ture techniques, eqn (9) is rewritten in terms of Lagrangian polynomials of parametric variable 5 over the interval [-1, 1] instead of [0, I]. Let {u(l; f)} denote the two component vector of w and 8. Then eqn (9) takes the form:

{q(t)]’ = [w,,w2,.

. . .

wp+,, et, e2,. . . , e,l.

(ii)

The nodal configurations for this hierarchy of virgin elements are illustrated in Table 2 for the cases p = 1,2,3. Nodal configurations for p > 3 can be inferred by induction. For p odd, the additional w dof located at 5 = 0 appears logical, attractive and symmetric. For p even, it may be placed anywhere, however, there is an advantage for locating it at a Gaussian point. Stiffness and consistent mass matrices as well as consistent load vectors for any order virgin element can be obtained by means of variational principle (5). The usual finite element methodology is involved. Gaussian quadrature with a minimum order necessary for exact integration is to be used.

(8) CONSTRAINED ELEMENTS

However, b,,, is left arbitrary and the Kirchhoff constraint cannot be enforced continuously throughout the length of the element. The coefficients b,,, are related to the nodal dof; e.g. for p = 1, the two node element with [0,, &] rotational dof, bz = 0, - 02 and for p = 2, the three node element with [e,, e2, &I, b, = 20, -4&-t 2&. There is no assurance that the linear combinations of & will result in b2 or ba being identically equal to zero. This simple argument reveals a cause for the ill-conditioning in the shear strain. It immediately suggests that w., and 0 should bear the same order interpolation, or w must begin with a polynomial one order higher than 0. Therefore, a general form for this interdependent variable interpolation is: P+l

w=

c

a.$"; n=O

8=n=O 2 b&“.

Elements in Refs. [ 10,111 used same order interpolations for both w and 0, and they cannot admit a continuous Kirchhoff constraint throughout. Their elements if applied to very slender beams would manifest an ill-conditioned shear strain. Nickel1 and Secor [ 131did use form (9) with p = 2 as their basis for their TIM7 element, but they required greater than CO continuity of w. Interdependent variable interpolations were used by Herrmann[ 17,181 in analyses of incompressible solids and plate bending using mixed variational principles. There has been an effort toward the use of discrete Kirchhoff constraints (at Gaussian points) in some nonconforming elements. Their convergence were verified by patch tests. See Gallagher[19] for a discussion of these formulations.

A continuous shear constraint condition may be used to generate a series of constrained elements from the virgin elements. The essence of this technique is the reduction of the polynomial order in the shear angle variation along the element. To develop this condition, take eqns (1) and (9) and express y as: ~=(a,-

bo)t(2a2-b1)&t...t((p

t l)a,+,-

b,)tP. (12)

Observe that lower orders of shear angle variation may be achieved by sequentially imposing constraints b, = (p + l)aP+l, b,-1 = pa,, etc. This constraint may be written as a differential shear relation:

L = [L,lI%J = 0

(13)

[&I = $

(14)

where

and yq denotes y whose highest order term in the polynomial is of order q. Note that 5’ (or x”) in eqn (12) is eliminated with q = p in eqn (13). After this term is gone, then yq = yP--l and repetition of eqn (13) removes the next order polynomial in the shear angle variation. Successively lower order shear angles are obtained by this process. To implement this constraint with displacement field (lo), y is written as

with

Virgin Elements

Elements employing interpolation field (9) will be termed virgin elements. In ‘anticipation of Gaussian quadra-

[LJ=[-$-l]=[;$-l].

(16)

338

A. TESSLER and S. B. DONG Table 2. Nodal configurations of conforming Timoshenko-type beam elements Constrained Elements

T3VC9

9

3

T3CQ8

8

3

T3CLl

7

2

6

1

,-, T3CC6 *x=i

0

Legend of Nodal Degrees of Freedom

l=

w;e

o=w

x -0

With eqns (IS) and (16), the differential shear relation

(13) takes the form

x3 = LILIbG

t)l= (y[ (f) -$T, - $1”‘”

a (17)

Substituting displacement field (10) into eqn (17) gives: ~~9= (2)‘[ i (;) g

Nw(L t), - $N.(&

t)]{s(r)} = 0. (18)

To eliminate a particular nodal dof, rearrange {q(t)} in eqn (18) so that the retained dof {q,} reside in the upper rows and let the deleted dof {qd}be in the lower position. The partitioned form of rearranged eqn (18) appears as: (19) Solving eqn (19) for {qd}yields: {qd) = [Bdl-‘[B,l{q,).

(20)

With eqn (20), the original nodal dof {q(t)} can be related to the retained nodal dot {q,} by the transformation:

{s(t)}= [Wqr~= [ ~Bd~!~B,l]~~r~.

interpolation field (22). It is again emphasized that Gaussian quadrature with a minimum order necessary for exact integration is to be used. Note that a virgin element of order p allows for the possibility of p constrained elements. For p = 1, only one constrained element can be generated with the internal deflection dof deleted. (see Table 2-the element designation used therein is explained in the next section). For p 2 2, eqn (22) represents the constraint equation for removal of the highest order polynomial variation in the shear angle. For successively greater reduction of the shear angle variation, eqns (17)~(22)must be repeated. In Table 2 are shown the constrained elements’ nodal configurations for p = 2 and p = 3 where the internal deflection dof have been eliminated. This method provides compatible shape functions for all constrained elements. For the stiffness matrix, the shear constraint is equivalent to static condensation of the same internal nodal dof. However, the present method effects the elimination a priori to the construction of the element matrices and load vectors. Static condensation, by contrast, involves a post-formulation deletion of the nodal dof. Moreover, it is applicable only to the stiffness and cannot be used for the mass matrix. With the shear constraint method, consistent mass matrices and load vectors are formulated directly from the appropriate interpolation functions for any order. No subsequent deletions of nodal dof are needed.

(21) ELEMENTDESIGNATION

Using eqn (21), displacement field (10) under the shear constraint with the highest order polynomial shear angle variation removed takes the form:

In order to refer to a particular beam element, the following four index label will be adopted: Tpbcn where T denotes Timoshenko and p-order p of the interdependent variable interpolation.

{u(f;

t)}

=

(22) b-type of element; b = u (virgin) or b = c (constrained),

The stiffness and consistent mass and consistent load vectors can be constructed in the usual manner using

c-order of the shear angle variation; c = C (constant), c = L (linear, c = Q (quadratic), etc. n-number of nodal dof in the element.

339

On a hierarchy of conforming Timoshenko beam elements

Examples of this labeling system are: TlVLS-virgin element with p = I, linear variation of y and 5 nodal dof. TlCU-constrained element with p = 1, constant y and 4 nodal dof. TZCLbconstrained element with p = 2, linear y and 6 nodal dof. TZCCS-constrained element with p = 2, constant y and 5 nodal dof. NUMERICAL EXAMPLES

Examples are presented to illustrate the static and dynamic behavior of certain elements in the present hierarchy. Those considered include the TIVLS, TlCC4, T2CL6 and T2CC5 elements. Reissner’s value of k* = 5/6 was employed throughout as the shear correction factor. 1. Static analysis Two cantilevered beams of rectangular cross-section were considered. One was moderately thick (L/H = 4), and the other was thin (L/H = 1000). An extensional/shear stiffness ratio E/G = 2.6 was used. Three loading conditions were analyzed: (1) concentrated tip load, (2) uniform load and (3) triangular load (see Fig. 1). On Fig. 2 are plots of the normalized tip deflection (normalization with respect to the exact Timoshenko value designated by wT as a function of element sub-

L Loading

I. Tip

load

Lx

_I

Sectith

X-X

conditions

2. Uniform

load

3. Triangular

lood

Fig. I. Cantilevered beam with rectangular cross-section.

A Uniform load IJ Triangular load

T2CL6 -I-

Number

2. Free vibrations To assess the inertial properties of this element hierarchy, natural vibrations of a simply-supported beam were considered. An extensional/shear stiffness ratio E/G = 2.667 was used because this value afforded an opportunity for direct comparison with previous results contained in Refs. [5,11]. Density p was chosen such that m = 1. Slenderness ratios L/r = lo,50 were considered, where r is the cross-sectional radius of gyration. For meaningful comparisons, the total degrees of freedom (dof) was limited to 40 or less. This led to the following dof for each of the element types: Ele.

No. of Ele.

dof

TlCC4 T2CL6 TlVL5 T2CC5

20 10 13 13

40 40 39 39

The frequencies for the lowest 10 modes are presented in Tables 3 and 4 for the cases L/r = 10,50, respectively. Also shown on these figures are the data of Archer (Ref. [5], 20 elements, 40 dof) and Dong and Wolf (Ref. [ll], 10 elements, 40 dof). In general, the results with the present hierarchy of elements compared favorably, with the T2CL6 element giving the best results. For the case L/r = 10, the present elements appear to model the second branch of Timoshenko modes much better, especially the pure shear mode. This is attributable to the continuous nature of the shear angle interpolation. In the case L/r = 50, the performance of the present elements deteriorated except for the T2CL6 element, which compared consistently better than the results of Refs. [5,11].

T2CC5 COMMENTS ON ELEMENTS BY REDUCED INTEGRATION TECHNIQUES

Triangular

2

divisions. It is observed that the first order elements converge quite rapidly. More impressive are the second order elements, which gave exact tip deflections for all discretizations (even with one element) and all load cases. Note that for the same order, both virgin and constrained elements led to identical results. This can be explained by recalling that the shear constraint condition in static analysis is equivalent to static condensation, i.e. elements of the same order are statically equivalent. In Fig. 3, the moment and shear force in the element nearest to the fixed end are illustrated. Both first and second order elements gave the same accuracy for the two cases of I./H considered. The exact values obtained in the point load case (even using one element only) is attributable to the fact that the assumed kinematic field used in the element derivation in fact corresponded to the exact solution. The results for uniform and triangular loadings showed relatively rapid convergence with mesh refinement.

4

0

load

16

of elements

Fig. 2. Normalized tip deflections of statically loaded cantilevered beam.

An interesting observation is that the stiffness matrices for a subset of constrained elements with one deflection dof removed from its virgin element (i.e. TlCC4, T2CL6, T3CQ8) and those based on selective reduced integration are identical. For example, that given by Hughes et al. [ I] (hereinafter called the HTK element) is the same as that for TlCC4, even though their deflection interpolations were different. This coincidence can be explained as

A. TESSLER and S. B. DONG

340

TIVLS,

LIH=4, TICC4,

Point load 0 Uniform load 0 Triangular load A

1000 T2CL6,

T2CC5

riangular

load

A

8 Number Fig.

3. Percentage error of moment and shear at center of element nearest to fixed end.

follows. Based on their respective interpolation fields, the shear a&es are: HTK:

y = ‘yot f(e, - Oz)[

TlCC4: where

yo

of elements

y = -yo

(2% (24)

is a constant given by yo= (w,- ~~$6-f(e,ted.

05)

The difference lies in the additional term $(&-&)g in eqn (23), and this term which describes the variation of the shear angle over the element may be questionable since it is exclusively in terms of the bending degrees of freedom. Fortuitously, with reduced integration of the shear energy using one-point Gaussian quadrature and sampling at 5 = 0, this term is eliminated. In the present derivation, all inte~ations are carried out using Gaussian quadrature with formulas sufficient for exact integration. Also, there is no ambiguity regarding the derivation of the consistent mass matrices and load vectors using the virgin or constrained interpolation polynomials, and this aspect was not addressed in Ref. [I]. Furthermore, with regards to stress recovery, the rule for reduced integration elements is to sample only at Gaussian intention points, which can easily be seen to annihilate the contribution of the questionable term. Such considerations need not be part of the methodology of the present hierarchy of elements. As further illustration, some numerical comparisons between the HTK and TlCC4 elements were carried out. As both their stiffnesses are identity, these ~omp~sons pertain to the nature of the load vectors and mass matrices. The ratio E/G = 2.667 was used. For a cantilevered beam under a static triangular load (see Fig. 1: load case 3), two methods for generating the load vectors were tried in connection with the HTK displacement interpolation, viz. (1) under integration (one-point Gaussian quadrature), and (2) exact integration. No suggestion is made that Hughes et af. advocated either of

these approaches; the motivation here is a curiosity in the comparison. Also, a consistent load vector using the TlCC4 constrained displacement field and a fumped load vector were employed. The results are summarized in Fig. 4. Observe that the TICC4 element gives the bending rotations exactiy for all values of L/r. One clear signal of nonconformability in the HTK methodology is convergence of the strain energy front above. In contrast, the consistent load vector of the TlCC4 element and the lumped load vector yielded the proper convergence of the strain energy, i.e. from befow (see Ref. [20: p. 361). Frequency results using mass matrices derived by various procedures are shown in Fig. 5 and Table 5. The so-called exact integrated mass matrix for the HTK element refers to exact integration of the kinetic energy using linear displacement and rotation polynomials. Again, no suggestion is made that Hughes et al. advocated exact inte~ation. The consistent mass matrix refers to the use of the TlCC4 constrained displacement field in the construction of the kinetic energy. A lumped mass matrix was also used. In Table 5 are shown the frequencies of the lowest four natural modes for various L./r ratios using a model composed of eight elements. By and large, frequencies using the consistent mass matrix compared better than the other forms of inertial representation. This was decidedly evident for the higher modes and for L/r large. In Fig. 5 are shown the convergence characteristics of the first mode for different L./r ratios as a function of dis~reti~tion. It is seen that except for one point (lumped mass case and L/r = lo), the results using the consistent mass were closer to the exact values, especially for the coarser discret~tions. CONCLUDINGREMARKS

A hierarchy of virgin and constrained Timoshenko beam elements has been presented. The interdependent inte~lations used herein and the meth~ology allow for the direct formulation of consistent element matrices. Moreover, conformability requirements in finite element theory are met.

.I

_

.”

i

_

_

^.

.”

^

-

-

Y

II

_

_

_

a

n

a= n

-a

T n

,_

_

-.

. .

^

“_ -. ._ * _ - -

I

b - second

x 100

a - pure shear mode

tEz

^

)El/cAL4)

-

_

_-

“”

*_

^

_

,” .”

-

L/r = 10

modes

1'2,

set of Timoshenko

t w, = a”

-

)

Table 3. Frequencies(I” for vibration of uniform hingedbeams

*r

=

”

aT

T a” - % x

n

“‘,

5.65 7.07

6.36 477.06 a.34 377.09 6.36 664.35

480.27 3M.74 669.96

3.28

8.64

4.38

3.13 294.23 4.S7 312.28

382.97

2.32

1.52

293.11

2.02 213.98

213.35

00

I

=

TlVLS

L/r

50

707.30

600.36

495.90

396.34

303.57

219.36

145.69 211.90

* 4.91

12.3) 696.37

11.2) 583.98

141.85

3.36

+

83.173

( 39 d.

T2CC5

2.w

I

a n

)

XC

( 40 d.o.f. ) I ( 39 d.o. !. 1

TICC4

1.20 143.10

I

I

~EI/PAL’~

an for vibration of uniform hinged beams

142.65

( 60 d.0.f.

1

=a n

Ref 1111

(w

Table 4. Frequencies

I ( 40 d.o.f. )

T2CL6

I

343

On a hierarchy of conforming Timoshenko beam elements

Table 5. Frequencies an for vibration of uniform hinged beams (on = an IEI/PAL'I 'I21

Slenderness Ratio

node

Timoshenko /Sol;;on

NTber

HTK

[I]

I

Lumped .\IassMotri:

TlCC4

I

L/r

10

50

1

8.36487

8.41742

0.63

2

25.1965

25.A25'

2.50

3

43.7749

47.8945

9.41

42.6933

-2.49

46.2623

5.68

4'

55.9016

55.9017

0.00

55.9017

0.00

55.9017

0.00

1

9.78902

9.97850

1.94

9.85109

0.63

9.84932

0.62

2

30.2444

41.2419

7.84

39.1769

2.44

39.0703

2.16

3

82.9863

97.9011

18.0

07.2489

5.14

86.1518

3.81

4

140.954

186.958

32.6

152.651

8.30

147.263

4.40

1

9.86966

10.0628

1.96

9.93428

0.65

2 3

88.8251 39.4781

;::;ii;

iif;

4

157.909

3464

an - a T NC=

n

1 ;zpE;

;;c;

x 100

aT

a

= pure shear mode

All examples were carried out with an R element model.

L/r=lO,

3464

!-e-e-Number

Load vectors

HTK CII : A - Underintegrated TICC4 : 0 -Consistent Lumped : x

1

of

elements 0 = Exactly

integrated

Fig. 4. Normalized tip deflections, rotations and strain energies of cantilevered beam with triangular load.

344

A. TESSLERand S. B. DONG

simple and efficientelement for plate bending. ht. J. Num.

Number

of elements

Mass matrices HTK Cl1 0 (Exactly integrated)

TlCC4 0 (Consistent)

Lumped

X

Fig. 5. Percentage error of fundamental frequency for uniform hinged beam.

The numerical examples revealed the high accuracy possible with these elements in static and vibrational problems over a wide range of beam thicknesses and frequencies. Although all elements of the first and second order are suitable for modeling purposes, the T2CL6 element gave exceptional overall performance. An explanation is given on the coincidence of the stiffness matrix derived by an ad hoc selective reduced integration with a subset of the constrained elements. This discussion clarified whatever ambiguities that may have existed on the construction of load vectors and mass matrices. The present methodology points toward a clear cut and direct formulation of all Timoshenko beam elements. REFERENCES 1.

T. J. R. Hughes, R. L. Taylor and W. Kanoknukulchai, A

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