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A discussion on “a hybrid, finite element-finite difference approach to the simplified large deflection analysis of structures by rudolph szilard[1]”
A DISCUSSION ON “A HYBRID, FINITE ELEMENT-FINITE DIFFERENCE APPROACH TO THE SIMPLIFIED LARGE DEFLECTION ANALYSIS OF STRUCTURES BY RUDOLPH SZILARD [ 11” GANGANPRATHAP Structures Division, National Aeronautical Laboratory, Bangalore, India (Received 23 March 1979; received forpublication 24 July 1979)
A new and considerably simplified technique for geometrically non-linear problems was proposed in [l]. The key to the simplification lies in two crucial assumptions that take liberties with the physics of the problem and describes a different physical system but produces a result fortuitously close to the correct answer at low non-linearities. We shall examine the two assumptions critically in what follows, for a beam problem. The same arguments hold true for a plate problem. 1.NEGLECTlONOF ExTENsIONAL Sl’RAINENERGYDUETO INPLANEDISPLACEMENT The large deformation of a beam can be described by two displacement quantities, the axial displacement, U, and the lateral displacement, w, of a point on the neutral axis. The axial strain due to stretching will then be
and the non-linear strain energy due to stretching will be U,,
=f
= AE( u., + ; w :)’ dx.
Ic
(2)
In [l] Szilard proposes that since u, is negligible in comparison with the rotation term w.,, the strain energy due to stretching can be replaced by UNL = ;
L AEwf, dx.
I0
(3)
This considerably simplifies the problem, in that inplane terms are eliminated from the functional for total potential and therefore, the beam problem can be studied using the familiar linear theory beam elements which have only terms in w and their dierivatives and do not
have degrees of freedom in the II direction. Actually, u,, is of the same order of magnitude as wfi and therefore cannot be neglected in the analysis Further, it is the deformation in the u direction that character&s large deflection problems, and this plays a sign&ant role in the analysis. A beam problem in which the non-linear stretching energy is defined by eqn (3) as
in Szilard, represents an artificial system in which every point in the beam is constrained to move vertically downwards, even while undergoing large deflection. This is not true of a realistic system. Very similar assumptions have been the basis of several finite element solutions to non-linear beam and plate problems[2-71 and this error has been pointed out in [8,9]. The system thus studied does not allow any in-plane displacement (i.e. u = 0 everywhere is implied) and hence is stiffer than the actual system whose non-linear strain energy is represented by eqn (2). This will easily be seen if a simple analysis is carried out using non-linear energies represented by eqns (2) and (3) respectively. 2. PRINCIPLE OF CONSERVATIONOF ENERGY TO ESTABLISH EQUILIBRIUM
A study of Szilard’s results for the test problem of a simply-supported beam subjected to a uniformly distributed load indicate that his results were less hardening than that obtained by a numerically exact analysis in [9] whereas the physical model studied by Szilard is stiffer than the actual problem modelled in [9]. The explanation for this lies in the manner in which the principle of conservation of energy is used in Szilard’s analysis, to establish equilibrium. In a simple one-term approximation, to a linear system, where the strain energy is a quadratic function of the displacement and the loaddisplacement relationship is linear, we can define a potential V due to the load, and a work W due to the load where W = (l/2) V, and a strain energy U due to deformation, and obtain the equilibrium position either, (a) by equating total strain energy to external work u=w or (b) by stating it as the stationary value of total potential S(U-
v)=o.
However, in a non-linear problem, the work W due to the load is not equal to (l/2) V because of the non-linear load-displacement relationship, and equilibrium is more 251
252
GANGANPRATHAP
simply established(because W is difficultto compute as the non-linear load-displacement relationship is not known a p&n’), using the stationary principle. S(U-
where
V)=O.
Szilard assumes that at moderately large deflections, the load-displacementrelationshipcan be approximated by a straightline and therefore W is taken as (l/2) V, and equilibriumis established by equating the strain energy to this work. We can see that the effect of this is to underestimatethe non-linearstrain energy by half, where the non-linear strain energy is a quartic function of displacement.Thus in a one-term solution, we can see that an extraneous factor of l/2 multipliesthe non-linear part of the solution, thereby reducing the non-linearity effect by half. We shall demonstrate this in a simple example below. The compoundingof the two errors produces a result that appears to be close to the actual values at low values of non-linearity. It is appropriate to include a note on what is meant by accuracy at low values of non-linearity. Since most of the approximate or simplifiedmethods of non-linear analysis involve approximationsmade in the non-linear strain energy, e.g. Rao et al. Mei, Szilard or even the Berger method[lO], a true estimate of the accuracy of this approximationwould be obtained only by comparingthe non-linear components of the results. At low non-linearities,obviously all approximationswill tend to the linear limiting values and a comparison of results may be misleading.We shall demonstrate this for the one term solution that follows.
An exact linear solutiongives a factor 0.01302instead of 0.01307,giving an idea of the accuracy of the one term solutionfor the linear part of the load displacement. CASE2
The simplifiedstructural model used by Mei, Rao and Raju and Szilard is now studied, again using the variational principleand a one term approximation,i.e.
=wf&+~ 2 -[P,,wdx)=O and this gives a load-displacementrelationship (3+4*5($=X Thus, this structural model is stiffer than the actual structure, the non-linear component being 50% higher than the actual value, and therefore an error of 50% is introduced due to the approximation. If however, the error is obtained by comparing the displacements obtainedfor a load X, we see that it is vanishinglysmall for displacementstendingto zero.
AONETERMENERGYSOLUTIONTOATfM'PROBL&I
CASE3
A uniform beam, hinged on immovable supports and carrying a uniformly distributed load is taken for study. First, we can establish that for such a problem, the inplaneequation gives
The simplifiedstructural model is now studied using the Szilard hypothesis of approximatingthe loaddisplacementrelation as a linear one. Then, the equationfor conservation of energy yields
1 N u,, t - w: = - = constant 2 EA where N is the axial force generated due to displacement, and it is simpleto show that w:,dx--
1 wfx. 2
and this gives a load-displacementrelationship.
Clearly, (u,,) is of the same order as wfx. CASE1
The actual, unsimplifiedstructure is studied applying the variationalprincipleto the total potential
Fig. l(a). Beam simply-supported on immovable hinges.
-[Powdx)=O and a one term approximation w = a sin (m/L). It can be
I-
L---__+
I
shown that the load-displacementrelationship for rectangularcross section beam is (;)+3(;)1=x
Fig. l(b). Simply-supported beam-one hinge movable.
A hybrid, finite element-finite difference approach to the simplified large deflection analysis of structures
253
2.0
($+2.25($=x.
(6)
Equation (6) is now less stiff than eqn (5) by a factor, l/2, and the non-linear component is in error, when compared to the correct answer, eqn (4), by 25%. When eqn (6) is plotted againstthe results of Szilard (Fig. 5 of [l]) the curves coincide almost exactly, showing that a one-term approximationof the Szilardmodel is nearly as accurate as the hybrid finite element-finite difference solution to his model. It is also interestingto note here that in the analysis of
I.5
I.0
-SJLILARD(Il x ----EON.4
Mei and Rao et al., a similar factor of l/2 is introduced
due to a linearisationprocess and this was pointed out in 191.
-
The same argumentsapply to the plate problem in [i], i.e. a correct analysis must include the effect of the in-planedisplacementquantities u and w. What requires a word of caution is that simplifiedmodels can easily be set up by neglectingsome of the non-linearstretchingof energy terms, so that the non-linear equations become uncoupled or become quasi-linearisedor allow a description usingreduced degrees of freedom. This is indeed the basis of models reported in [l-7] and the several dozen or more papers based on the Berger method[lO]. All of these methods may give fortuitously close answers where overall results are concerned (e.g. load-displacement curves, frequency-amplitude of vibration curves, etc.) but may considerablyunderestimateor overestimate the non-linear stretching energy terms and the inplane stress patterns. A case in point is the Berger approximationand this was pointed out in [ 11,121.
REFERENCES
1. R. Szilard, A hybrid, finite element-finite differenceapproach
to simplifiedlarge deflectionanalysis of structures. Compvt. Structures 9,341-350 (1978). 2. C. Mei, Finite element displacement method for large amplitude free flexural vibrations of beams and plates. Comput. Structures 3, 163-174(1972). 3. C. Mei, Non-linear vibrations of beams by matrix displacement method. AIAA J. 10,355-375 (1972).
--EON.6
0 0
CONCLUSIONS
EON.6
0.5
IO
5
I5
20
x----r
Fig. 2. Non-dimensional load-deflection relationship.
4. C. Mei, A finite-element approach for non-linear panel flutter. AIAA J. 15.1107-1110(1977). 5 G. V. Rae; K. K. ~R$u and I. S. Raju, Finite element formulation for the large amplitude free vibrations of beams and orthotropic circular plates. Comput. Structures 6, 169 172(1976). 6. G. V. Rao, I. S. Raju and K. K. Raju, A finite element formulation for the large amplitude flexural vibration of rectangular plates. Comput. Structures 6, 163-167(1976). 7. K. K. Raju, G. V. Rao and I. S. Raju, Effect of geometric non-linearity on the free flexural vibrations of moderately thick rectangular plates. Comput. Structures 9, 441444 (1978). 8. G. Prathap, Comment on “A finite element approach for non-linear panelflutter”. AIAA J. 16, 863864 (1978). 9. G. Prathap, Comments on “Effects of lingitudinal or inplane deformation and inertia on the large amplitude flexural vibrations of slender beams and thin plates”. J. Sound Vib. 55,308-311 (1977). 10. G. Prathap, T. K. Varadan and L. S. V. B. Rao, Non-linear analysis of simply supported beams (to be published). II. H. M. Berger, A new approach to the analysis of large deflections of plates. J. Appl. Mech. 22,464-472 (1955). 12. G. Prathap, The Berger approximation-A critical reexamination. J. Sound Vib. To be published.
A new and considerably simplified technique for geometrically non-linear problems was proposed in [l]. The key to the simplification lies in two crucial assumptions that take liberties with the physics of the problem and describes a different physical system but produces a result fortuitously close to the correct answer at low non-linearities. We shall examine the two assumptions critically in what follows, for a beam problem. The same arguments hold true for a plate problem. 1.NEGLECTlONOF ExTENsIONAL Sl’RAINENERGYDUETO INPLANEDISPLACEMENT The large deformation of a beam can be described by two displacement quantities, the axial displacement, U, and the lateral displacement, w, of a point on the neutral axis. The axial strain due to stretching will then be
and the non-linear strain energy due to stretching will be U,,
=f
= AE( u., + ; w :)’ dx.
Ic
(2)
In [l] Szilard proposes that since u, is negligible in comparison with the rotation term w.,, the strain energy due to stretching can be replaced by UNL = ;
L AEwf, dx.
I0
(3)
This considerably simplifies the problem, in that inplane terms are eliminated from the functional for total potential and therefore, the beam problem can be studied using the familiar linear theory beam elements which have only terms in w and their dierivatives and do not
have degrees of freedom in the II direction. Actually, u,, is of the same order of magnitude as wfi and therefore cannot be neglected in the analysis Further, it is the deformation in the u direction that character&s large deflection problems, and this plays a sign&ant role in the analysis. A beam problem in which the non-linear stretching energy is defined by eqn (3) as
in Szilard, represents an artificial system in which every point in the beam is constrained to move vertically downwards, even while undergoing large deflection. This is not true of a realistic system. Very similar assumptions have been the basis of several finite element solutions to non-linear beam and plate problems[2-71 and this error has been pointed out in [8,9]. The system thus studied does not allow any in-plane displacement (i.e. u = 0 everywhere is implied) and hence is stiffer than the actual system whose non-linear strain energy is represented by eqn (2). This will easily be seen if a simple analysis is carried out using non-linear energies represented by eqns (2) and (3) respectively. 2. PRINCIPLE OF CONSERVATIONOF ENERGY TO ESTABLISH EQUILIBRIUM
A study of Szilard’s results for the test problem of a simply-supported beam subjected to a uniformly distributed load indicate that his results were less hardening than that obtained by a numerically exact analysis in [9] whereas the physical model studied by Szilard is stiffer than the actual problem modelled in [9]. The explanation for this lies in the manner in which the principle of conservation of energy is used in Szilard’s analysis, to establish equilibrium. In a simple one-term approximation, to a linear system, where the strain energy is a quadratic function of the displacement and the loaddisplacement relationship is linear, we can define a potential V due to the load, and a work W due to the load where W = (l/2) V, and a strain energy U due to deformation, and obtain the equilibrium position either, (a) by equating total strain energy to external work u=w or (b) by stating it as the stationary value of total potential S(U-
v)=o.
However, in a non-linear problem, the work W due to the load is not equal to (l/2) V because of the non-linear load-displacement relationship, and equilibrium is more 251
252
GANGANPRATHAP
simply established(because W is difficultto compute as the non-linear load-displacement relationship is not known a p&n’), using the stationary principle. S(U-
where
V)=O.
Szilard assumes that at moderately large deflections, the load-displacementrelationshipcan be approximated by a straightline and therefore W is taken as (l/2) V, and equilibriumis established by equating the strain energy to this work. We can see that the effect of this is to underestimatethe non-linearstrain energy by half, where the non-linear strain energy is a quartic function of displacement.Thus in a one-term solution, we can see that an extraneous factor of l/2 multipliesthe non-linear part of the solution, thereby reducing the non-linearity effect by half. We shall demonstrate this in a simple example below. The compoundingof the two errors produces a result that appears to be close to the actual values at low values of non-linearity. It is appropriate to include a note on what is meant by accuracy at low values of non-linearity. Since most of the approximate or simplifiedmethods of non-linear analysis involve approximationsmade in the non-linear strain energy, e.g. Rao et al. Mei, Szilard or even the Berger method[lO], a true estimate of the accuracy of this approximationwould be obtained only by comparingthe non-linear components of the results. At low non-linearities,obviously all approximationswill tend to the linear limiting values and a comparison of results may be misleading.We shall demonstrate this for the one term solution that follows.
An exact linear solutiongives a factor 0.01302instead of 0.01307,giving an idea of the accuracy of the one term solutionfor the linear part of the load displacement. CASE2
The simplifiedstructural model used by Mei, Rao and Raju and Szilard is now studied, again using the variational principleand a one term approximation,i.e.
=wf&+~ 2 -[P,,wdx)=O and this gives a load-displacementrelationship (3+4*5($=X Thus, this structural model is stiffer than the actual structure, the non-linear component being 50% higher than the actual value, and therefore an error of 50% is introduced due to the approximation. If however, the error is obtained by comparing the displacements obtainedfor a load X, we see that it is vanishinglysmall for displacementstendingto zero.
AONETERMENERGYSOLUTIONTOATfM'PROBL&I
CASE3
A uniform beam, hinged on immovable supports and carrying a uniformly distributed load is taken for study. First, we can establish that for such a problem, the inplaneequation gives
The simplifiedstructural model is now studied using the Szilard hypothesis of approximatingthe loaddisplacementrelation as a linear one. Then, the equationfor conservation of energy yields
1 N u,, t - w: = - = constant 2 EA where N is the axial force generated due to displacement, and it is simpleto show that w:,dx--
1 wfx. 2
and this gives a load-displacementrelationship.
Clearly, (u,,) is of the same order as wfx. CASE1
The actual, unsimplifiedstructure is studied applying the variationalprincipleto the total potential
Fig. l(a). Beam simply-supported on immovable hinges.
-[Powdx)=O and a one term approximation w = a sin (m/L). It can be
I-
L---__+
I
shown that the load-displacementrelationship for rectangularcross section beam is (;)+3(;)1=x
Fig. l(b). Simply-supported beam-one hinge movable.
A hybrid, finite element-finite difference approach to the simplified large deflection analysis of structures
253
2.0
($+2.25($=x.
(6)
Equation (6) is now less stiff than eqn (5) by a factor, l/2, and the non-linear component is in error, when compared to the correct answer, eqn (4), by 25%. When eqn (6) is plotted againstthe results of Szilard (Fig. 5 of [l]) the curves coincide almost exactly, showing that a one-term approximationof the Szilardmodel is nearly as accurate as the hybrid finite element-finite difference solution to his model. It is also interestingto note here that in the analysis of
I.5
I.0
-SJLILARD(Il x ----EON.4
Mei and Rao et al., a similar factor of l/2 is introduced
due to a linearisationprocess and this was pointed out in 191.
-
The same argumentsapply to the plate problem in [i], i.e. a correct analysis must include the effect of the in-planedisplacementquantities u and w. What requires a word of caution is that simplifiedmodels can easily be set up by neglectingsome of the non-linearstretchingof energy terms, so that the non-linear equations become uncoupled or become quasi-linearisedor allow a description usingreduced degrees of freedom. This is indeed the basis of models reported in [l-7] and the several dozen or more papers based on the Berger method[lO]. All of these methods may give fortuitously close answers where overall results are concerned (e.g. load-displacement curves, frequency-amplitude of vibration curves, etc.) but may considerablyunderestimateor overestimate the non-linear stretching energy terms and the inplane stress patterns. A case in point is the Berger approximationand this was pointed out in [ 11,121.
REFERENCES
1. R. Szilard, A hybrid, finite element-finite differenceapproach
to simplifiedlarge deflectionanalysis of structures. Compvt. Structures 9,341-350 (1978). 2. C. Mei, Finite element displacement method for large amplitude free flexural vibrations of beams and plates. Comput. Structures 3, 163-174(1972). 3. C. Mei, Non-linear vibrations of beams by matrix displacement method. AIAA J. 10,355-375 (1972).
--EON.6
0 0
CONCLUSIONS
EON.6
0.5
IO
5
I5
20
x----r
Fig. 2. Non-dimensional load-deflection relationship.
4. C. Mei, A finite-element approach for non-linear panel flutter. AIAA J. 15.1107-1110(1977). 5 G. V. Rae; K. K. ~R$u and I. S. Raju, Finite element formulation for the large amplitude free vibrations of beams and orthotropic circular plates. Comput. Structures 6, 169 172(1976). 6. G. V. Rao, I. S. Raju and K. K. Raju, A finite element formulation for the large amplitude flexural vibration of rectangular plates. Comput. Structures 6, 163-167(1976). 7. K. K. Raju, G. V. Rao and I. S. Raju, Effect of geometric non-linearity on the free flexural vibrations of moderately thick rectangular plates. Comput. Structures 9, 441444 (1978). 8. G. Prathap, Comment on “A finite element approach for non-linear panelflutter”. AIAA J. 16, 863864 (1978). 9. G. Prathap, Comments on “Effects of lingitudinal or inplane deformation and inertia on the large amplitude flexural vibrations of slender beams and thin plates”. J. Sound Vib. 55,308-311 (1977). 10. G. Prathap, T. K. Varadan and L. S. V. B. Rao, Non-linear analysis of simply supported beams (to be published). II. H. M. Berger, A new approach to the analysis of large deflections of plates. J. Appl. Mech. 22,464-472 (1955). 12. G. Prathap, The Berger approximation-A critical reexamination. J. Sound Vib. To be published.