compurrrr & srrrMmres Vol. 21. No. I, pp. 83-U. 1986

Printed !n Great Britam.

DISCUSSION OF FINITE ELEMENT FORMULATIONS OF NONLINEAR BEAM VIBRATIONS

CHUH MEI Department of Mechanical Engineering and Mechanics, Old Dominion University. Sorfolk,

Virginia 23508

Abstract--The Lagrange-type, Galerkin, and Ritz-type finite element formulations for large amplitude vibrations of immovably supported slender beams are reexamined. Inconsistency in the definition of frequency or criterion of defining nonlinearity is discussed, and validity of the frequency solution is examined. Improved finite element results by including both longitudinal displacement and inertia in the formulation are presented and compared with available Rayleigh-Ritz continuum solutions.

I. INTRODUCTIOS

Nonlinear free flexural vibrations of beams with ends restrained from longitudinal movement have

been investigated using the classical continuum ap- proach[l-41. In obtaining the governing equation of motion

the longitudinal inertia I(,,, was neglected. The av- erage tensile force due to deflection is then ap- proximated by

where E, I, m, A, and L are the Young’s modulus, moment of inertia of the cross-sectional area, mass per unit length, area of cross-section, and length of the beam, respectively. The exact solution of eqn (1) for a simply supported (7, 14, and 18) of Ref. [l])

w(x, t) = cc sin

= a sin

and

beam is given by (eqns

F F(t)

y cn(pr, k) (3)

(z)‘= ($)‘[I +$(;)‘I, (4)

where k, p, and K are the modulus, circular fre- quency, and quarter period of the elliptic function, and p, wo, and w are the radius of gyration of cross- sectional area, linear frequency, and nonlinear fre- quency, respectively. From eqns (3) and (4), the tensile force due to the deflection alone iS

N = + ; ’ cn’(pt, k), 0

and it varies between zero and the maximum value of

N P,, cl 2

max = - - 7

0 4 P

where P,, = 7’EI/L2 is the Euler buckling load. Evensen[3] has obtained solutions for beams with various support conditions, and Ray and Bert[4] have used the assumed time mode approach. An excellent survey of the literature on nonlinear anal- yses of beams concerning classical continuum methods is given by Sathyamoorthy[S].

The finite element methods[6-91 have also been developed for nonlinear free vibrations of immov- ably supported beams. In the earlier Ritz-type finite element formulation[6, 71, an average axial force is assumed for each beam element instead of the entire beam in the continuum approach. To account for the variation of tensile force N between zero and N max in a linearized sense, a factor of one-half is employed for the nonlinear stiffness matrix. This is not a.n error to reduce the nonlinear stiffness matrix to half its value, as commented in Ref. [IO]. Rao et a1.[8, 91 neglected the longitudinal displacement II and linearized the nonlinear strain-displacement re- lation in their formulation. These earlier Ritz-type finite element results[6-91 all give excellent agree- ment with the continuum solutions[l-41 (also see Table 1). Sathyamoorthy[l l] has also given a com- prehensive review on finite elements for nonlinear static and dynamic analysis of beam structures.

2. DISCUSSIOXS OF A XEW SONLISEARITY CRITERIOS

Prathap and Varadan[lZ] presented a new deli- nition for frequency as an alternative criterion to describe the degree of nonlinearity. Their new fre- quency is evaluated at the instant (I = r,) of max- imum amplitude (F(f,) = 1, dF(t, )ldr = 0 and d2F(r,)ldt’ = -w’F at the point of reversal of mo- tion). This creates confusion by comparing fre-

83

8-t CHCH .vEI

quency results (Fig. 2 of Ref. [12]) based on two different definitions. the conventional[ l-4. 6-91 and the new[ 10. 121 frequency criteria.

Bhashyam and Prathap[ 131 and Sarma and Var- adan[l4] recently presented a Galerkin and a La- grange-type finite element formulation for nonlin- ear beam vibrations based on the so-called new nonlinearity criterion. Some of the results given in Refs. [13, 141 need clarification. Firstly, compari- sion of finite element frequency results based on the conventional (as an example. the fourth column in Table 2 of Ref. [14]) and the new (the third, fifth. and sixth columns of Table 2[ 141) criteria is mean- ingless. Also, the classical continuum frequency data (w/w,,)’ given in Tables 3-5 of Ref. [ 131 and in Tables 2-4 of Ref. [ 141 do not agree with the results given originally by Woinowsky-Krieger, Burgreen, and Evensen[ l-31.

claimed to be the exact assumed space mode so- lution based on the new criterion. In fact. the fre- quency-amplitude relation of eqn (7) does not rep- resent any real physical problem.

Further examination raises some questions about the validity of this new criterion of defining nonlinearity. The variable stretching force N of eqn (2) in the governing equation of motion, eqn (1). is treated as a constant by Sarma and Varadan (see eqns (10, 14) of Ref. [IO] and eqns (6, 7, and 9) of Ref. [141) and Bhashyam and Prathap (see eqns (7, 8) of Ref. [13]). What they really solved, based on the classical definition, is basically a linear vibra- tion problem subjected to an initial axial tensile force which was investigated by Lurie[lS]. In ad- dition, Prathap et nl.[ 10, 12-141 unrealistically eval- uated this constant (which should be a variable. as shown in eqn (5)) tensile force at the point of rev- ersal of motion which they claimed to be a new nonlinearity criterion. This leads to a frequency ratio[lS] for a simply supported beam as

w ( > 2 N - = 1 +a

WO PC,

=I,! a2

0 4 P . (7)

This is exactly what Prathap and Varadan (eqn (17) of Ref. [12]) and Bhashyam and Prathap (see foot- note in Table 3 of Ref. [13]) have obtained and

3. IMPROVED RITZ-TYPE FORJlL’L.ATIOS

Most recently, an improved Ritz-type finite ele- ment formulation for nonlinear free flexural vibra- tions of beams has been developed[l61. The im- provement is to include both longitudinal deformation and inertia (LDI) in the formulation. The displacement functions chosen for the beam element are in the simplest form

11 = fl5 + 06.T. (8)

The element nodal displacements at the two end nodes are

where the subscripts h and M denote bending and membrane components, respectively. The kinetic and strain energies of the beam element are given

by

(10)

where I is the length of the beam element. Again, using the usual finite element procedure,

the stiffness equation of motion governing the non- linear beam vibration problem is given by

[LYb pnm] ($1 + ([ib k] + p, F_]) {S} = 0. (12)

Table I. Free vibration frequency ratios o/w” for a simply supported beam with immovable axial ends

Without LDP With LDI (Lip = 100)

Elliptic New Finite element[6-81 Rayleigh Finite element]l61

function criterion Ritz

_ solution n result First Final solution First Final

P [I, 21 [IO. II, 141 iteration result [I71 iteration result

1.0 I .0892 1.1180 I .089S I .0888 I .0607 1.0613 1.0613(3?

2.0 1.3178 1.4142 I .3203 I.3119 I .2246 I .I’70 I .X69(4)

3.0 I .6257 I .8028 I .6295 I .60X I .4.573 I .4620 I .4617l4)

4.0 I .9760 2.2361 1.9761 I.9216 I .7309 I .7383 I .7375(63

5.0 2.3501 1.6926 2.3396 2.2544 2.0289 1.0393 2.0378(7)

a Longitudinal deformation and inertia. b Number in parenthesis denotes the number of iterations to _eet a converged

solution.

Discussion of finite element formulations of nonlinear beam vibrations 85 -

where [ml. [X-l. and [k] denote the element consis- tent mass, linear stiffness, and nonlinear stiffness matrices, respectively. Derivation of these matrices

are given in [ 161. The fundamental frequency ratios o/w0 at various amplitude a/p for a simply supported beam of slenderness ratio Lip = 100 are shown in Table 1. Only one-half of the beam modelled with six equal elements was used. Earlier finite element results without LDI[6-81 and the improved finite element results]161 are both given in the table. It shows that the effect of including LDI in the for- mulation is to reduce the nonlinearity. The contin- uum solution of eqn (4)[ 1, 21 is also given to dem- onstrate the closeness with the earlier finite element (specially the first iteration) results. This is because both the continuum and the first iteration finite ele- ment employed a linear mode shape in the analysis. Raju et a/.[171 used the Rayleigh-Ritz method in investigation of the effects of LDI on large ampli- tude flexural vibrations of slender beams. The ap- propriate frequency-amplitude relationship using Rayleigh-Ritz method is also given in Table 1. This clearly demonstrates the remarkable agreement be- tween the improved Ritz-type finite element and Rayleigh-Ritz solutions. The Lagrange-type finite element results[ 141 are also given to show the poor agreement.

-1. CONCLUDISG REMARKS

The inconsistency and validity of the new fre- quency criterion of the Galerkin and the Lagrange- type finite element formulations are investigated. An improved Ritz-type finite element method, by including longitudinal displacement and inertia in the formulation, is presented. The frequency results obtained agree very well with the Rayleigh-Ritz so- lutions.

REFERENCES

1. S. Woinowsky-Krieger, The effect of an axial force on the vibrations of hinged bars. J. Appl. Mech. Trmns ASME 17, 35-36 (1950).

2. D. Burgreen. Free vibrations of a pin-ended column with constant distance between pin ends. J. Appl. Mech. Trans ASME 18, 135-139 (1951).

3. D. A. Evensen, Sonlinear vibrations of beams with various boundary conditions. AMA J. 6, 370-372 (1968).

4.

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- ,

C. Slei, Nonlinear vibrations of beams by matrix dis- nlacement method. AlAA J. 10, 355-357 (1972). c. Mei, Finite element displacement method for large amplitude free oscillations of beams and plates. Com- plrr. Strrrcrures 3, 163-174 (1973). G. V. Rao. K. Kanake Raju, and I. S. Raju. Finite element formulation for the large amplitude free vi- brations of beams and orthotropic plates. Cornput. Srrucrures 6, 169-172 (1976). G. V. Rao, I. S. Raju, and K. Kanaka Raju. Sonlinear vibrations of beams considering shear deformation and rotary inertia. AIAA J. 14, 685-687 (1976). B. S. Sarma and T. K. Varadan. Certain discussions in the finite element formulation of nonlinear vibration analysis. Comput. Srrucrures 15, 643-646 C 1982). M. Sathyamoorthy, Nonlinear analysis of beams part II: Finite element methods. Shock Vibr. Dig. 14, 7- 18 (1982). G. Prathap and T. K. Varadan. The large amplitude vibrations of hinged beams. Comput. Srrwrures 9, 219-222 (1978). G. R. Bhashyam and G. Prathap. Galerkin finite ele- ment method for nonlinear beam vibrations. J. Sound Vibr. 72, 191-203 (1980). B. S. Sarma and T. K. Varadan, Lagrange-type for- mulation for finite element analysis of nonlinear beam vibrations. J. Sound Vibr. 86, 61-70 (1983). H. Lurie, Lateral vibrations as related to structural stability. J. Appl. Mech. Trans ASME 19, 195-204 (1952). C. .Mei and K. Decha-Umphai, A finite element method for nonlinear forced vibrations of beams. Proc. 2nd Intl. Conf. on Recent Advances in Strcrc- turn/ Dynamics. Institute Sound and Vib. Research, Univ. of Southampton, 319-328 (1984). I. S. Raju, G. V. Rao, and K. Kanaka Raju. Effect of longitudinal or inplane deformation and inertia on the large amplitude flexural vibrations of slender beams and thin plates. J. Sound Vibr. 49, 415-422 (1976).