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Curved finite elements for the analysis of shallow and deep arches
Con~prtcrs & Sfrac~ares, Vol. 4, pp. 559-580.
CURVED
Pergamon Press 1974. Printed in Great Britain
FINITE ELEMENTS FOR THE ANALYSIS SHALLOW AND DEEP ARCHES
OF
D. J. DAWE Department
of Civil Engineering,
University of Birmingham,
Birmingham,
England.
l
Abstract-The paper considers the analysis of shallow and deep arches by curved beam finite elements, the interest in this problem being that it forms a limiting case of the shell problem. The individual elements are assumed to be shallow with respect to a local base line and different types of strain-displacement equations are utilised. Particular attention is focussed on the choice of the displacement patterns for the elements and models based both on independently-interpolated displacement components, of up to quintic order, and on assumed strain distributions are considered. Numerical applications using the various models show that of those tested only quinticquintic models and assumed-strain models are generally satisfactory.
1. INTRODUCTION IN RECENT years a great deal of effort has been concentrated on the development of suitable finite element models for the analysis of thin curved structures. Various approaches of radically different character have been adopted but this paper is concerned only with the basis for curved displacement models based on thin arch or shell theory and the Kirchoff normalcy condition. Attention has naturally been concentrated on shell analysis but there is considerable virtue in studying the simpler limiting case of the curved beam so as to gain insight into the shell problem. Of course, some of the difficulties associated with a shell analysis, such as those concerning the displacement compatability requirements, are not a significant feature of the arch analysis but in other respects a study of the simpler structure can yield valuable information which tends to be hidden in shell models by the complexity of the analysis. The literature on curved shell finite elements based on two-dimensional shell theory is large (see, for example, a review paper by the author [ 11)although comparatively few studies deal with shells of an arbitrary geometry. For the analysis of such shells one of several different avenues of approach are possible. Firstly standard general shell equations written in terms of curvilinear coordinates can be directly used but this results in a complicated tensorial formulation. Alternatively the shell equations and, in particular, the expression for the strain energy can be re-written in terms of Cartesian coordinates and used in conjunction with assumed Cartesian components of displacement. Another and more commonly used approach, is to adopt the simplified theory obtained by employing shallowness assumptions and to assemble an arbitrary deep structure from the resulting shallow elements. It is particularly with this latter type of approach that this paper is concerned, though the implications of the work are of a more general nature. Shell elements based on shallow theory have been proposed for the solution both of shells of a truly shallow global geometry and of shells which might be globally deep so long as the individual elements are locally shallow. A common failing of the majority of the ‘Formerly at Central ElectricityGenerating Board, Berkeley Nuclear Laboratories, Berkeley, Gloucestershire. 559
560
D. J. DAWE
latter category of analyses has been the assumption of the truly-shallow strain-displacement equations in the calculation of the strain energy of an element advocated for use in the solution of deep problems. In fact, the simplified expressions for rotation and curvature in the shallow theory need to be extended by terms involving the tangential displacement components if deep problems are to be accommodated; this is confirmed by the numerical results presented in this paper. An important feature of the arch (or shell) problem and one which influences the choice of the assumed element displacement pattern is the coupling that exists between the components of displacement in the definitions of strains and curvatures, rigid-body motion, etc. In this paper a number of curved beam element models are considered in which the displacements are assumed to be either independently interpolated as (relatively) highorder polynomials or are coupled so as to represent simple states of strain and curvature. The use of independently-interpolated displacement components is the more common approach and is also the more straightforward in concept. However, more degrees-offreedom are often needed to adequately represent strain-free and strain-inducing states with this sort of model than with a coupled displacement model. The additional degrees-offreedom used here are those higher displacement derivatives at the end nodes whose continuity is not strictly required by the variational formulation. Amongst general shell elements which are based on high-order independently-interpolated displacement components are those of Cowper et al. [2, 31, Argyris and Scharpf [4] and Dupuis and Go&l [S]; in [2 and 31 cubic polynomials are assumed for the membrane components and a reduced quintic polynomial for the normal component whilst in [4 and 51 quintic polynomials are used for all (Cartesian) displacement components. The assumption of a coupled displacement field to precisely represent both rigid-body motions and simple, independent strain states is an attractive alternative approach. The author used such an approach in a Marguerre-type analysis of a truly-shallow arch [6] and this resulted in a very efficient model using only those degrees of freedom needed to satisfy continuity requirements. Subsequent application of the same philosophy by Ashwell et al. [7] to the generation of a circular arch element has also demonstrated the validity of this approach. The assumed strain approach can yield an element with excellent convergence characteristics but a disadvantage is that the derivation is not always simple. 2. STRAIN-DISPLACEMENT
EQUATIONS
The strain-displacement equations for the arbitrary shell due to Koiter [8] reduce to the following form for the arbitrary curved beam shown in Fig. 1:
1 &=A
du w Z--R’
(14
1
dw u da+R’
(lb)
4=z
and
UC)
Curved Finite Elements for the Analysis of Shallow and Deep Arches
Shl
Here E is the extensional strain of the neutral axis, 4 the rotation of the normal and X the change of curvature; u and w are the tangential and normal components of displacement, R the initial radius of curvature, a the curvilinear coordinate, and A the parameter such that Ada is the line element along ct. In terms of the Cartesian coordinates x and z the equations (1) can be written (the dash denotes differentiation with respect to X)
t=u’cose-i2
R’
c$= w’cod + ;,
(24
W)
and
,=,,d$(w’cose+~) where O=tan-’ (z’) and l/R=& beams of any geometry.
(24
cos 0. These equations are applicable to thin curved
FIG. 1. Arch geometry.
The basis of the usual shallow arch theory resides in assumptions not only of the shallowness of the initial geometry (z, the height of the arch above the base line, is small enough such that z’* < 1, l/R-z”, cos 6~ 1) but also of the magnitude of the tangential displacement relative to the transverse displacement; the tangential displacement is omitted from the expressions for rotation and additional curvature. These assumptions result in the simplified strain-displacement equations which correspond to the equations of Vlasov [9] for the shallow shell,
562
D. J. DAWE &EU’ - M’Z”,
(34
rp=w’,
(3b)
x= w”.
(3c)
and
The neglect of the tangential displacement in the expressions for rotation and additional curvature is justifiable for the truly-shallow structure but not so, in general, for a globallydeep structure (as will be confirmed numerically in a later section). If such neglect is not made and the shallowness assumptions only are used a further set of approximate equations results, &=d - wz”,
W
d)=w’+uz”,
(4b)
and x=u”’ + u’z” + uz”‘.
(W
These simplified equations will be suitable for the finite element analysis of globallydeep arches provided that the individual elements are locally shallow. The above equations are formulated in terms of the tangential and normal components of displacement, u and w. Alternately, the equations may be recast in terms of displacement components, ii and 1, measured in the Cartesian directions. Then the equations for the arbitrary arch become
E=COS2e(ii’ + W’z’), c#F-cos26(i-vii’z’),
W
(5b)
and - G’z’]). ~=cos%(E” - ii”Z’ - ll’z” - 2Z’Z”COS%[W’
(5c)
For the truly-shallow arch these equations reduce to E=ii’+W’Z’,
(6b) and (6~) and correspond to the Marguerre strain-displacement equations for shallow shells [lo]. For the locally-shallow arch the relevant equations are taken to be
f+#-_11’z’,
(7b)
and %+Y”_ $‘z‘ - i’z” - 2z’z”#.
(7c)
Curved Finite Elements for the Analysis of Shallow and Deep Arches
563
Note that the last term in the curvature expression is small and could reasonably be neglected. It is included here, though, because its presence results in zero additional curvature under a true rigid-body motion, as will be seen in the next section. 3. STRAIN-FREE
AND STRAIN-INDUCING
STATES
The displacement pattern assumed for the curved finite element should satisfy the requirements of kinematic admissibility. Here this means that the tangential and normal components of displacement together with the rotation should be continuous across element boundaries. In addition, a finite element which is proposed for general application must be able to adequately accommodate both strain-free states and strain-inducing states (particularly the simple strain states) if it is to have good convergence properties. 3.1 Strtlin-free
states
For the arbitrary curved beam shown in Fig. 1 the normal and tangential displacement components of the exact rigid-body motion can be shown, by methods presented elsewhere [111, to be u,=cos&?, + sin&J, + cos@z - xz’)wY
(ga)
and wI= - sin0& + cos&5, -
00se(x+
ZZ’)C+.
@b)
Here S,, 6, are the components of a rigid-body translation and w, is the rigid-body rotation. The rigid-body motion is consistent (i.e. gives zero strains) with the strain-displacement equations (1) but not, of course, with the approximate strain-displacement equations of the preceding section. Reduced rigid-body motions can be obtained from the exact ones by using similar assumptions as were employed in obtaining the reduced strain-displacement equations. Proceeding along these lines leads to the displacement fields l/,=6,+
z’6, +(z-xz’)o,
(94
and 1v,=6z-Xxog
(9b)
for the truly-shallow arch element, together with II,= 6, + z’6, + (z - XZ’)OY
(lOa)
and II‘,= - z’bx + 6, -x0,
(lob)
for the locally-shallow arch. The motions (9) yield vanishing strain and curvature in equations (3) whilst the motions (10) yield finite (but small) values of strain and curvature for the locally-shallow arch (equations (4)). Where an element has constant initial curvature-or, more precisely, constant z”it is possible and more convenient to express the strain-free motions for the truly-shallow and for the locally-shallow arch in another form. For the former arch the motions are
and
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D. J. DAWE
where a, etc. are constants. For the locally-shallow arch the homogeneous displacement equations are identically satisfied by the motions tl,=a, + (Izsin(z”x) - ascos(z”x)
strain-
(12a)
and w,=azcos(z”x)
+ azsin(z”x).
(12b)
In the alternative formulation where the displacement components are in the Cartesian directions the exact rigid-body motion is
(134 and (13b) and this is unchanged by any shallowness assumptions. This rigid-body motion precisely satisfies the homogeneous strain-displacement equations for the arbitrary arch, the trulyshallow arch and the locally-shallow arch. (For the latter arch this would not be so were the final term of the additional curvature expression of equation (7~) not included; then a very small curvature would result under the rigid-body motion.) 3.2 Basic strain states In this section the displacement patterns corresponding to linearly-varying states of strain and curvature in some of the curved arch elements are presented. These strain patterns are obtained by integrating the assumed strain and curvature variations e=a.++a,x
UW
x=a6 +a+
(14b)
and
where a4 . . . a, are undetermined constants and E and x are defined in terms of the arch displacements and their derivatives as in Section 2. Considering first the arbitrary truly-shallow arch where the arch behaviour is governed by the strain-displacement equations (3), the displacements corresponding to the above strain pattern are
us=
f$x2zr- 1:Ixz’dx)
+ %x3zf-Ji
3x’l’dx)
+a4x+as<
(Isa)
and w,=
a6
a7
-x2 + - x3. 2
6
(15b)
Where the arch has constant initial curvature the tangential displacement can be more conveniently written +a&)
+a,x+asg.
U5c)
Curved Finite Elementsfor the Analysisof Shallowand Deep Arches
565
For the analysis of the arbitrary shallow arch based on Cartesian components of displacement and the strain-displacement equations (6) the displacement components are
and
x2
i3s=a6-2
+a,;.
x3
WI
For the locally-shallow arch of constant initial curvature the displacements corresponding to the assumed strain pattern and the strain-displacement equations (4) are
and
(17b)
ws=
A solution (involving logarithmic and inverse trigonometric functions) can also be found for the locally-shallow arch with Cartesian displacement components and the straindisplacement equations (7). However, this solution has not been incorporated into any finite element model since it will be shown that use of the strain-displacement equations (6) gives adequate results for the deep problems considered.
4. ARCH ELEMENTS In modelling the geometry of the unloaded arch by elements of the Marguerre type the height of the arch, z(x), above the local base line is assumed to be a cubic polynomial whose coefficients are equated to the height and slope at the two end nodes. On the other hand, in using elements of the Vlasov type in the circular arch applications described in Section 5 it is more convenient to input the curvature directly. 4.1 Elements for shallow structures It can be shown that one of the two Euler equations of the shallow curved arch variational problem indicates that the extensional strain is uniform in an interval between loads. This is so whether the arch strain energy is based on the Vlasov strain-displacement equations (3) or the Marguerre equations (6). Thus there is no point in assuming anything higher than a uniform value of extensional strain in a truly-shallow arch element. For the truly-shallow element it has been seen that the displacement components corresponding to strain-free and to uniform-strain, linear-curvature states are of polynomial form whose order depends on the order of the expression for the height z(x). The membrane displacement component corresponding to such states contains higher-order terms than the normal component and this suggests that the usual procedure of adopting a lowerorder independent polynomial expression for the membrane component than for the
D. J. DAWE
566
normal component is not the most efficient one. If the height of the arch centre line above the base line is taken to be a cubic function of x then the independent membrane component of displacement should be of quintic order if it is deemed necessary to accurately represent a state of uniform strain. Eight element models have been developed for truly-shallow arch analysis and with these models the local and global axes coincide. Five of the models (designated Sl to S5) are based on the Vlasov strain-displacement equations (3) whilst the remaining three (S6 to S8) are based on the Marguerre equations (6). The eight elements are as follows:Sl. A ‘quintic-quintic’ element with the independent components of displacement
(184 and (18b) The nodal degrees-of-freedom are U, a’, u”, w, w’ and w” and all are made continuous at the nodes (this implies continuity of rotation, additional curvature and extensional strain in the shallow arch). S2. A ‘quintic-cubic’ element with the components of displacement
+A,X+A3x2+&3
u=A,
(194
and w=Ag+Asx+A,X2fAsX3+AgX4+AIOX5. The nodal degrees-of-freedom the nodes.
(lob)
are U, u’, w, w’, W” and all are made continuous
at
S3. A ‘cubic-cubic’ element with components of displacement
and
The nodal degrees-of-freedom nodes.
are U, u’, w, w’ and are all made continuous at the
S4. A ‘condensed cubic-cubic’ element with components of displacement defined as equation (20) but with the ‘extra’ degree-of-freedom U’ at each node condensed out by minimising the element potential energy, assuming that the potential energy of the loads The final degrees of freedom are thus is independent of the extra degree-of-freedom. only the ones necessary for kinematic compatibility, i.e. U, w, NJ’. S5. A constant-strain, linear-curvature element based on the coupled components of displacement (assembled from equations (11 and 15) with a,=0 in the latter equation),
x2
u=A,,+A,x+z"
A3x+A4y+A5
x3
3+A,g
x4
(214
and w=A3+A4x+A,x2+A,x3.
@lb)
Curved Finite Elements for the Analysis of Shallow and Deep Arches
567
The degrees-of-freedom are again u, w, w’. 5’6. A quintic-quintic element with the Cartesian components of displacement fi and - --I --I, - --I E defined as in equation (18) and with the nodal degrees-of-freedom u, u , II , w, w and ,E”, S7. A cubic-cubic element with the Cartesian components of displacement E and Z - -_I defined as in equation (20) and with the nodal degrees-of-freedom u, u , w and is’. S8. The original constant-strain, linear-curvature element of Ref. [6] with the nodal degrees-of-freedom ii, 9, a’, based on the displacement components (obtained from equations (13 and 16) with a,=O), E=A,+A,x-Ap2A,(
xz-
{;zdx)
-3&(
x’z$Zxrdx)
(224
and iC=A,+A,x+A,x2+A,x3.
(22b)
4.2 Elements for deep structures As with the truly-shallow arch element, models for deep structures are considered which are based on independently-interpolated displacement components or on coupled components describing simple strain states, including the zero-strain state. It is noted that the nature of the strain states is quite different from that associated with the trulyshallow arch (see Section 3) and that the corresponding displacement components are not always of polynomial form. Also, in contrast to the shallow arch, the extensional strain for the deep structure is clearly not uniform, in general, in the interval between loads but this need not mean that an element restricted to a uniform distribution of membrane strain is inefficient. In analysing deep arches the end points of an element lie on the arch centre-line and the local Cartesian axes obviously do not coincide with the global axes. The question arises as to what global degrees-of-freedom should be used. One possibility is to take as degrees-of-freedom the tangential and normal components of displacement together with derivatives of these quantities with respect to the curvilinear coordinate in the arch middle surface. However, connection of such derivatives at the node points means in general that the values of rotation, extensional strain and additional curvature, calculated from the approximate equations given earlier, cannot match. Although, the magnitude of such mis-match will be small (since the individual elements are shallow) it appears more appropriate to directly connect at the nodes quantities such as the (approximate) rotation, strain and curvature as required. The elements considered for the analysis of deep arches are as follows:Dl. A quintioquintic element based on the deep strain-displacement equations (4) (it follows that the assumed components of displacement are the surface ones u and w). The global degrees-of-freedom are u, E, E’, w, 4 and x. 02. A quint&cubic element based on equations (4) and with degrees-of-freedom u, E, w, 4 and x. 03. A cubic-cubic element based on equations (4) and with degrees-of-freedom u, E, w and 4. 04. A condensed cubic-cubic element again based on equations (4) and with degreesof-freedom u, w and 4.
568
D. J. DAWE
D5. Two variations of an assumed-strain model based on equations (4) and the displacement components (from equations (12 and 17)). +A,x +TASX’Z”
u =A, +&sin(z”x)-A,cos(z”xj
W-4
and ~=A~cos(z”~)+A~sin(z”x)+A,
+ A+
W)
In the first approach A, is put equal to A5 to give a constant-strain, linear-curvature element with degrees-of-freedom u, w and 4 at the end nodes. In a second approach AS and A, are independent of one another and the seven coefficients of equation (23) are equated to u, w and 4 at the element end points together with the tangential displacement at the element centre. This latter degree-of-freedom is eliminated by static condensation. 06. A quintic-quintic element with assumed Cartesian components of displacement ii and R. This model is, like element S6, based on the strain-displacement equations (6). The global degrees-of-freedom are U, E, E’, w, 4 and x. 07. A cubic-cubic element with assumed Cartesian components of displacements and with global degrees-of-freedom U, E, w and 4. Variations of this model derived from both shallow and deep expression for the rotation and curvature (equations (6 and 7)) have been considered; these variations are discussed in Section 5. 08. A constant-strain, linear-curvature element as model SS (i.e. based on equations (6) and the assumed displacement components as equation (22)) but with the transformation to global degrees-of-freedom U, w and 4. D9. A constant-strain, linear-curvature element as (shallow) model S5 but used in a deep context with global degrees-of-freedom U, w and 4.
5. NUMERICAL
STUDIES
The models described in the last section have been applied to the solution of a number of arch problems using a standard finite element programme employing double-precision arithmetic throughout. The forces and bending moments are calculated in the usual manner via displacement derivatives. 5.1 Shallow arch The arch is circular with a base length of 34 in. and a central rise of 1.09 in. (corresponding to a radius of 133.1138 in.): it carries a central vertical load and is clamped at its ends. Two arch thicknesses of l/16 in. and 1 in. are considered. In the finite element analysis a symmetric half of the arch is discretised into elements of equal base length whose local axes coincide with the arch global axes. For the thinner arch Fig. 2 illustrates the convergence with mesh refinement of the deflection, w,, under the central load (as a percentage of the exact value) for all the trulyshallow elements. It is noted that due to the approximations inherent in the different shallow strain-displacement equations, the results obtained using Marguerre-type elements (S6-S8) coverge to slightly different values than do the Vlasov type elements (Sl-S5) and that both types of converged solutiondiffer to a small extent from the exact solution. Figure 2 shows that the two quintic-quintic elements, $1 and S6 are very efficient in this application.
Curved Finite Elements for the Analysis of Shallow and Deep Arches
569
When using model Sl the same central deflection, to 6 significant figures, is obtained for all element assemblages from one to 16 elements in the half-arch. When using model S6 there is a small error (0.21 per cent) in the single-element solution but convergence is very rapid; the error is a positive one because of the geometric approximations involved in modelling the true arch. Apart from these elements the two uniform-strain linearcurvature elements converge most rapidly and have a very similar performance. It should be noted that, although differing on a degree-of-freedom basis the results for the Vlasov cubic-cubic element (S3) and the reduced cubic-cubic element (S4) are very similar for the same number of elements. 100 -
1
I
Y
0
I 20
I IO FINAL
FIG. 2. Convergence
DEGREES
I 30 OF FREEDOM.
0
of central deflection for shallow thii arch.
For the thicker arch convergence of the central deffection is very rapid for all element types and there is so little difference between the various models that graphical presentation of the results is superfluous. It is noted that the relative accuracy demonstrated by the variousmodelsin the thin arch analysis is unchanged and that with all models the solutions obtained with only two elements in the half-arch are within 0.35 per cent of the corresponding converged solution. In particular, the single-element results for both quinticquintic models are effectively identical to the converged solutions. Having obtained a broad picture of the convergence characteristics of the various shallow elements using the central deflection as the measure of comparison it is instructive to examine the distribution throughout the arch of some relevant quantities. Fortunately, in view of the number of element types tested, it will not be necessary to present complete results corresponding to all models in order to demonstrate the salient features of the various models. So far as the distribution of displacement components is concerned it is noted that the normal deflection at the centre is a valid indicator of the accuracy of the overall distribution of the components and that these distributions are smooth with no apparent evidence of any significant oscillation.
570
D. J. DAWE
a.DISPLACEHENT
b. BENDING
COMPONENTS.
MOMENT, 1
c. FORCE. !! I
FIG. 3. Distributions using model S3 for shallow thin arch.
In Figs. 3-6 are shown the distributions of the displacement components, bending moment and force obtained using, in turn, the cubic-cubic model S3 and the quintic-cubic model S2 in the analysis of the thin and thick arch. In these (and subsequent) figures AB is the zero line, A being a clamped boundary and B the arch centre line. Note that the two displacement components are not drawn to the same scale, w being very much larger than u (in Figs. 3, 4, 5, 6) the scale of w is 50 (25) times that of u). It can be seen that the displacement distribution of the thin arch is less simple than that of the thick arch and this, in some part, contributes to the reduced accuracy of the results for the thin arch. A prominent feature of these results is the waviness of the force distributions. Severe oscillations of the force occur in analysing the thin arch with few elements. The magnitude of these oscillations reduces with increase in the number of elements and (markedly) with increase in the thickness of the arch. Some waviness is also apparent in the bending moment distribution obtained using the quint&cubic element. Similar oscillatory behaviour of finite element force and moment distributions has been noted by Cowper et al. [2] using a shallow shell model which is broadly a two-dimensional equivalent of the quintic-cubic arch model.
Curved Finite Elements for the Analysis of Shallow and Deep Arches
-
EXACT
---.
n-2
____
571
n=,
a. bISPLACEHENT COMPONENTS
t,.BENOlNG MOMENT,
--____c--c.
FIG. 1.
Distributions
FORCE.
using model S3 for shallow thick arch.
The distributions shown in Figs. 3 and 4 for the Vlasov-type cubic-cubic element S3 are quite similar to those obtained using the reduced cubic-cubic element S4 and the Marguerre-type, cubic-cubic element S7. For all these elements the bending moment distribution is formed of linear segments. Also, the oscillations in the force distribution are common though it is noted that the strain is highly discontinuous at the node points when using model S4 and that the waviness is more marked in using model S7 than in using model S3. The distributions of displacement, moment and force obtained using the quintic-quintic models for both the thin and thick arches are very accurate. With the Vlasov-type element, Sl, the results are virtually indistinguishable from the exact (deep) ones even when using only a single element in the half-arch. The oscillatory behaviour of the force distributions of the other independently-interpolated displacement models is effectively absent (some slight variation of the force is present but at most the waviness is of the order of one part in a million). With the Marguerre-type element, S6, the displacement and moment distribution are, again, virtually indistinguishable from the exact distributions but there is some small, though not insignificant, fluctuation in the force distribution. For the thin arch the range of these fluctuations is f 3.5 per cent using a single element but this reduces rapidly to less than + 0.1 per cent for three elements. For the thick arch the greatest fluctuation, using a single element, is less than 0.07 per cent.
572
D. J. DAWE
-
b.
BENDIWG
EXACT
MOMENT. I
FIG. 5. Distributions using model S? for shallow thin arch.
Finally, the uniform-strain models, S5 and 58, yield moment distributions which are very similar one to another and which differ little from the distributions shown in Figs. 3 and 4 for the Vlasov cubic--cubic element. Highly accurate force distributions are obtained using models S5 and S8; when two or more elements are used in the half-arch the (uniform) magnitude of the force is within half a per cent of the exact value for both the thick and thin arch. 5.2 Deep circular arches Geometrical details of these arches are as those given for the preceding shallow arch except that now the central rise is 17 in. (i.e. the arch is semi-circular). The arch again carries a central vertical load but, as well as the clamped boundary condition, a condition that the ends of the arch are free to slide horizontally in separately applied (this corresponds to the pinched ring). These problems are basically of an inextensional nature. Equal length elements are used in the analysis of the symmetric half-arch and models Dl-D9 are tested in these applications.
Curved Finite Elements for the Analysis of Shallow and Deep Arches
---.-
EXACT nm I
n-t
A:B a. DISPiACEHENl
b. BENDING
COMPONENTS.
MOMENT,
. -___-I-
c.
FORCE
FIG. 6. Distributions
using model S2 for shallow thick arch.
FtG. 7. Convergence of central deflection, thin clamped arch.
573
574
D. J. DAWE
The convergence of the central deflection with the number of elements in the half-arch, n, is shown in Figs. 7-10 for the clamped and the pinched arch, and for the two arch thicknesses (again & in. and 1 in., corresponding to R/t ratios of 272 and 17). With one exception all models shown ultimate convergence to effectively the same deflection in each particular case. As expected, model D9 which is based on shallow expressions of the Vlasov type for the rotation and additional curvature converges not to the true solution but to a level substantially below this. On the other hand the Marguerre-type models 06, 07 and D8 which are based on shallow rotation and curvature expressions do demonstrate convergence to the true, deep solution. As with the use of flat elements the reason for this is presumably due to the coupling effect al the nodes when the local Cartesian displacement components are linked. The Marguerre cubic-cubic model has in fact been tested in these problems with the curvature expression given in turn by equation (6c), by the full equation (7~) and by equation (7~) with the last term neglected. The differences between the three sets of results thus obtained are small and reduce with mesh refinement. The deflection results (leaving model D9 out of further consideration) fall broadly into two groups. Both types of quinticquintic element (Dl and 06) and of assumed-strain element (D5 and 08) yield consistently accurate results, whatever the arch thickness. The other lower-order, independently-interpolated models (02, 03, 04 and 07) are less successful and their accuracy is dependent on the thickness of the arch, being poorer for the thin arch. Of course the use of the approximate strain-displacement equations affects the accuracy of all models, particularly when few elements are used, and probably accounts for most of the error shown for models D1, D5, 06 and 08.
0
I 2
I
I
,
I
I
I
4
6n
a
IO
I2
FIG. 8. Convergence
of central deflection, thick clamped arch.
The quintic-quintic model results for a particular type of analysis are more accurate, on an element basis, than corresponding assumed-strain model results, though only marginally so for the pinched arch; it will be remembered, however, that approximately twice as many degrees-of-freedom are involved in the solutions using the quintic-quintic models. In the applications considered here the convergence of the quintic-quintic and of the
Curved Finite Elements for the Analysis of Shallow and Deep Arches
575
assumed-strain models based on the (shallow) Marguerre-type analysis is more rapid than that of the corresponding models based on the (deep) Vlasov-type analysis. With regard to the Vlasov-type, assumed-strain model D5 it is noted that the results obtained from the assumption of a uniform variation of extensional strain are virtually identical in these particular applications to those obtained from the assumption of a linear variation of strain. The quintiwubic element 02 offers no significant improvement, in these applications, over the corresponding (Vlasov) cubic-cubic elements, 03 and 04. It can be seen that the latter elements yield similar results one to another; relaxation of the continuity of the ‘extra’ degree-of-freedom, E, at the nodes makes little difference to the results. This is the expected result in applications where the loading is such that the strain is, in fact, everywhere continuous. Nevertheless this finding for elements based on the assumption of local shallowness is in direct contrast to that of Ashwell, Sabir and Roberts [7] in their work on circular arches where it is concluded that it is essential to relax the excess continuity condition in deep arch prob1ems.t
0
2
4
6
n
8
IO
I2
16
Fro. 9. Convergence of central deflection, thin pinched arch.
Before leaving the discussion of the displacements it is further noted that the wellknown simple flat element (cubic-linear) has also been used, for comparison purposes, in the analysis of the deep arches. In these applications the performance of the flat element is roughly similar to that of model D5 and is thus superior to all arch models other than those of the assumed-strain or quintic-quintic type.
t These authors have since shown that their conclusion is incorrect in a note (IN. J. Mech. Sci. 15, 325-327, 1973) appearing since the present paper was written.
576
D.
I 0
,,
2
4
FIG. 10. Convergence
o.HODEL
E. HODEL
J. DAWE
I
I
6n
’
I
I IO
I2
16
of central deflection, thick pinched arch.
D5
Dt
b.
HODEL
d. MODEL
DJ
DI
t. HODEL FIG.
11. Moment distributions,
thin clamped arch.
D6
-.-.
-
EXACT n - 2
-_---
n - 4 n - 6
Curved Finite Elements for the Analysis of Shallow and Deep Arches
577
The distribution of relevant quantities throughout a representative deep arch-the clamped one-will now be considered. Again, in this deep problem, the accuracy of the calculated central deflection is a good indicator of the accuracy of the overall distribution of the displacement components. The true distribution of the displacements, for both the thin and thick arches, is roughly of the same form as that illustrated for the shallow, thin arch in Fig. 3a. The bending moment and force distributions calculated using models Dl, 02, 03, D5 and 06 for the two deep, clamped arches are shown in Figs. 11-14. The results for the Marguerre uniform-strain element, 08,are very similar to those shown for model D5 (as indeed are the distributions based on the use of flat elements). The results shown as model D5 are for that version where a uniform strain distribution is assumed ; the linear-strain version yields effectively identical moment distributions and centreelement force values. Model 04 results are closely comparable to the illustrated distributions for model 03 except that the force is not single-valued at the nodes. The nature of the Marguerre cubic-cubic model 07 results is also similar to that of model 03.
-_-._
EXACT n2
----.._ ___.
n=, “-8 h=,6
a.MODEL
D5
c. MODEL
DZ
d.
MODEL
DI
e.
MODEL
D6
*lB
FIG. 12. Force distributions,
thin clamped arch.
The illustrated distributions of moment and force show again that the accuracy of the results obtained using the quintic-quintic models and the assumed-strain (and flat) models is good and is hardly affected by the relative thickness of the arch. Use of the quintic-quintic models yields a very accurate bending moment distribution. The dependence of the accuracy of the quintiocubic and cubic-cubic model results upon the relative thickness of the arch, noted earlier in considering the convergence of the central deflection, is accentuated in these moment and force distributions. For the thin arch the moment distribution calculated using these elements is poor and the fluctuations in the force distributions, using a practical number of elements, are many times greater than the true magnitude of the force. It is recalled, though, that since the problem is primarily a bending one the force is of secondary importance to the bending moment.
D. J. DAWE
a.HODEL
D5
b. MODEL
D3
c. MODEL
DZ
d. MODEL
DI
e. MODEL
FIG. 13. Moment distributions,
a. MODEL
05
D6
thick clamped arch.
b. MODEL
03
d. MODEL
DI
/
EXACT ---n.* --n-4 ____nrb
c. MODEL
D2
FIG. 14. Force distributions,
o. MODEL thick clamped arch.
06
--n-4 ____nx8
Curved Finite Elements for the Analysis of Shallow and Deep Arches
579
6. CONCLUSIONS The numerical results of a representative shallow-arch application show that quinticquintic models, based on either a Marguerre or a Vlasov type of analysis, give excellent results, for thick or thin geometry, even when only a single element is used. The success of these models was anticipated following a consideration of the governing equations of the problem and of the nature of the strain-free and simple straining states of the shallow curved beam element. The assumed-strain models considered also give reliable results though these are not as precise as the quinticquintic model results. The lower-order, independently-interpolated models (quintic-cubic and cubic-cubic) are characterised by the fact that their efficiency is dependent upon the relative thickness of the arch. Calculated force distributions based on these models show oscillatory behaviour which is very marked for a thin arch. No significant improvement in accuracy is achieved by increasing the order of the normal deflection component from cubic to quintic if the tangential component is not similarly increased. In fact it would probably be more efficient to increase the order of the tangential component rather than the normal one (in a cubic-quintic model, say) but this has not been pursued in the present study. The results obtained for the deep-arch applications using locally-shallow elements further emphasise the above findings. The models which successfully deal with both thick and thin deep arches are again the quint&quintic and assumed-strain models. The other models considered (quintic-cubic and cubic-cubic) give force and moment distributions of a very oscillatory nature for the thin arch and it follows that their shell equivalents would exhibit similar tendencies in like circumstances. In considering the shell problem the concept of the assumed-strain model is an attractive one since it requires the use of only a minimal number of degrees-of-freedom. Unfortunately the derivation of such models in a shell analysis of any generality is difficult and the displacement components corresponding to the assumed-strain pattern will, in general, not be compatible at element boundaries. On the other hand the application of independentlyinterpolated displacement fields in shell analysis is more straightforward in concept and reduces compatibility difficulties. The results presented here suggest that for the development of an efficient general shell element all three displacement components should be of essentially quintic order. Such a conclusion might possibly be unduly restrictive, however, since the present work has omitted consideration of any models based on displacement components of quartic order or, as mentioned earlier, of models having a tangential component of higher order than the normal one. The strain-displacement equations which have been used in the present study are approximate ones involving the assumption of shallowness with respect to a local base line. This source of error could be removed, at the expense of increased complexity, by using unreduced strain-displacement equations in a similar analysis and this might be particularly appropriate in shell work. It would not be expected that the conclusions regarding the relative merit of the various types of model would be changed. Consideration of the approximate equations has confirmed that the use of locally-shallow elements based on the standard (shallow) Vlasov strain-displacement equations leads to incorrect solution of deep problems. On the other hand, due to the coupling effect at the nodes on connecting adjacent elements, convergence to the true solution of deep problems can be obtained when using models based on the Marguerre shallow equations.
580 Acknowledgement-This
D. J. DAWE paper is published
by permission
of the Central Electricity Generating
Board.
REFERENCES [I] D. J. DAWE, Curved finite elements in the analysis of shell structures by the finite element method. Pm. 1st Znt. Conf. on Structural Mechanics in Reactor Technology, Vol. 5. Part J, Springer, Berlin (1971). [21 G. R. COWPER,G. M. LINDBERGand M. D. OLSON, A shallow shell finite element of triangular shape. /nt. J. Solids Struct. 6, 1133-l 156 (1970). Pi G. R. COWPER, G. M. LINDBERGand M. D. ORSON, Comparison of two high-precision triangular finite elements for arbitrary deep shells. Proc. 3rd Conf. on Matrix Methods in Structural Mechanics, Wright-Patterson Air For& Ba& Ohio (1971). t41J. H. ARGYRI~and D. W. SCHARPF,The SHEBA family of shell elements for the matrix displacement method. Aero. J. 72, 873-883 (1968). [51G. DUPUI~ and J. J. GOAL, A curved finite element for thin elastic shells. Znt. J. Solids Struct. 6, 1413-1428 (1970). El D. J. DARE, A finite deflection analysis of shallow arches by the discrete element method. Royal Aircraft Establishment (Famborough) Tech. Rep. 70058, April (1970); also published in Int. J. Num. Meth. Engqg 3,529-552 (1971). 171D. G. ASHWELL,A. B. SABCRand T. M. ROBERT&Further studies in the application of curved finite elements to circular arches. Int. J. Mech. Sci. 13,507-517 (1971). 181W. T. KOITER,A consistent first approximation in the general theory of thin elastic shells, Delft (1959). (North-Holland, 1960.) 1V. Z. VLASOV,General theory of shells and its application in engineering. NASA ‘ITF-99 (1964). _[9J [lo] K. MARGUERRE,Zur theorie der gekriimmten Platte grosser formtinderung. Proc. Znt. Co& Appl. Mech., 93-101 (1938). [ll] D. J. DAWE, Rigid body motions and strain-displacement equations of curved shell finite elements. Znt. J. Mech. Sci. 14, 569-578 (1972). (Received 6 March 1973)
CURVED
Pergamon Press 1974. Printed in Great Britain
FINITE ELEMENTS FOR THE ANALYSIS SHALLOW AND DEEP ARCHES
OF
D. J. DAWE Department
of Civil Engineering,
University of Birmingham,
Birmingham,
England.
l
Abstract-The paper considers the analysis of shallow and deep arches by curved beam finite elements, the interest in this problem being that it forms a limiting case of the shell problem. The individual elements are assumed to be shallow with respect to a local base line and different types of strain-displacement equations are utilised. Particular attention is focussed on the choice of the displacement patterns for the elements and models based both on independently-interpolated displacement components, of up to quintic order, and on assumed strain distributions are considered. Numerical applications using the various models show that of those tested only quinticquintic models and assumed-strain models are generally satisfactory.
1. INTRODUCTION IN RECENT years a great deal of effort has been concentrated on the development of suitable finite element models for the analysis of thin curved structures. Various approaches of radically different character have been adopted but this paper is concerned only with the basis for curved displacement models based on thin arch or shell theory and the Kirchoff normalcy condition. Attention has naturally been concentrated on shell analysis but there is considerable virtue in studying the simpler limiting case of the curved beam so as to gain insight into the shell problem. Of course, some of the difficulties associated with a shell analysis, such as those concerning the displacement compatability requirements, are not a significant feature of the arch analysis but in other respects a study of the simpler structure can yield valuable information which tends to be hidden in shell models by the complexity of the analysis. The literature on curved shell finite elements based on two-dimensional shell theory is large (see, for example, a review paper by the author [ 11)although comparatively few studies deal with shells of an arbitrary geometry. For the analysis of such shells one of several different avenues of approach are possible. Firstly standard general shell equations written in terms of curvilinear coordinates can be directly used but this results in a complicated tensorial formulation. Alternatively the shell equations and, in particular, the expression for the strain energy can be re-written in terms of Cartesian coordinates and used in conjunction with assumed Cartesian components of displacement. Another and more commonly used approach, is to adopt the simplified theory obtained by employing shallowness assumptions and to assemble an arbitrary deep structure from the resulting shallow elements. It is particularly with this latter type of approach that this paper is concerned, though the implications of the work are of a more general nature. Shell elements based on shallow theory have been proposed for the solution both of shells of a truly shallow global geometry and of shells which might be globally deep so long as the individual elements are locally shallow. A common failing of the majority of the ‘Formerly at Central ElectricityGenerating Board, Berkeley Nuclear Laboratories, Berkeley, Gloucestershire. 559
560
D. J. DAWE
latter category of analyses has been the assumption of the truly-shallow strain-displacement equations in the calculation of the strain energy of an element advocated for use in the solution of deep problems. In fact, the simplified expressions for rotation and curvature in the shallow theory need to be extended by terms involving the tangential displacement components if deep problems are to be accommodated; this is confirmed by the numerical results presented in this paper. An important feature of the arch (or shell) problem and one which influences the choice of the assumed element displacement pattern is the coupling that exists between the components of displacement in the definitions of strains and curvatures, rigid-body motion, etc. In this paper a number of curved beam element models are considered in which the displacements are assumed to be either independently interpolated as (relatively) highorder polynomials or are coupled so as to represent simple states of strain and curvature. The use of independently-interpolated displacement components is the more common approach and is also the more straightforward in concept. However, more degrees-offreedom are often needed to adequately represent strain-free and strain-inducing states with this sort of model than with a coupled displacement model. The additional degrees-offreedom used here are those higher displacement derivatives at the end nodes whose continuity is not strictly required by the variational formulation. Amongst general shell elements which are based on high-order independently-interpolated displacement components are those of Cowper et al. [2, 31, Argyris and Scharpf [4] and Dupuis and Go&l [S]; in [2 and 31 cubic polynomials are assumed for the membrane components and a reduced quintic polynomial for the normal component whilst in [4 and 51 quintic polynomials are used for all (Cartesian) displacement components. The assumption of a coupled displacement field to precisely represent both rigid-body motions and simple, independent strain states is an attractive alternative approach. The author used such an approach in a Marguerre-type analysis of a truly-shallow arch [6] and this resulted in a very efficient model using only those degrees of freedom needed to satisfy continuity requirements. Subsequent application of the same philosophy by Ashwell et al. [7] to the generation of a circular arch element has also demonstrated the validity of this approach. The assumed strain approach can yield an element with excellent convergence characteristics but a disadvantage is that the derivation is not always simple. 2. STRAIN-DISPLACEMENT
EQUATIONS
The strain-displacement equations for the arbitrary shell due to Koiter [8] reduce to the following form for the arbitrary curved beam shown in Fig. 1:
1 &=A
du w Z--R’
(14
1
dw u da+R’
(lb)
4=z
and
UC)
Curved Finite Elements for the Analysis of Shallow and Deep Arches
Shl
Here E is the extensional strain of the neutral axis, 4 the rotation of the normal and X the change of curvature; u and w are the tangential and normal components of displacement, R the initial radius of curvature, a the curvilinear coordinate, and A the parameter such that Ada is the line element along ct. In terms of the Cartesian coordinates x and z the equations (1) can be written (the dash denotes differentiation with respect to X)
t=u’cose-i2
R’
c$= w’cod + ;,
(24
W)
and
,=,,d$(w’cose+~) where O=tan-’ (z’) and l/R=& beams of any geometry.
(24
cos 0. These equations are applicable to thin curved
FIG. 1. Arch geometry.
The basis of the usual shallow arch theory resides in assumptions not only of the shallowness of the initial geometry (z, the height of the arch above the base line, is small enough such that z’* < 1, l/R-z”, cos 6~ 1) but also of the magnitude of the tangential displacement relative to the transverse displacement; the tangential displacement is omitted from the expressions for rotation and additional curvature. These assumptions result in the simplified strain-displacement equations which correspond to the equations of Vlasov [9] for the shallow shell,
562
D. J. DAWE &EU’ - M’Z”,
(34
rp=w’,
(3b)
x= w”.
(3c)
and
The neglect of the tangential displacement in the expressions for rotation and additional curvature is justifiable for the truly-shallow structure but not so, in general, for a globallydeep structure (as will be confirmed numerically in a later section). If such neglect is not made and the shallowness assumptions only are used a further set of approximate equations results, &=d - wz”,
W
d)=w’+uz”,
(4b)
and x=u”’ + u’z” + uz”‘.
(W
These simplified equations will be suitable for the finite element analysis of globallydeep arches provided that the individual elements are locally shallow. The above equations are formulated in terms of the tangential and normal components of displacement, u and w. Alternately, the equations may be recast in terms of displacement components, ii and 1, measured in the Cartesian directions. Then the equations for the arbitrary arch become
E=COS2e(ii’ + W’z’), c#F-cos26(i-vii’z’),
W
(5b)
and - G’z’]). ~=cos%(E” - ii”Z’ - ll’z” - 2Z’Z”COS%[W’
(5c)
For the truly-shallow arch these equations reduce to E=ii’+W’Z’,
(6b) and (6~) and correspond to the Marguerre strain-displacement equations for shallow shells [lo]. For the locally-shallow arch the relevant equations are taken to be
f+#-_11’z’,
(7b)
and %+Y”_ $‘z‘ - i’z” - 2z’z”#.
(7c)
Curved Finite Elements for the Analysis of Shallow and Deep Arches
563
Note that the last term in the curvature expression is small and could reasonably be neglected. It is included here, though, because its presence results in zero additional curvature under a true rigid-body motion, as will be seen in the next section. 3. STRAIN-FREE
AND STRAIN-INDUCING
STATES
The displacement pattern assumed for the curved finite element should satisfy the requirements of kinematic admissibility. Here this means that the tangential and normal components of displacement together with the rotation should be continuous across element boundaries. In addition, a finite element which is proposed for general application must be able to adequately accommodate both strain-free states and strain-inducing states (particularly the simple strain states) if it is to have good convergence properties. 3.1 Strtlin-free
states
For the arbitrary curved beam shown in Fig. 1 the normal and tangential displacement components of the exact rigid-body motion can be shown, by methods presented elsewhere [111, to be u,=cos&?, + sin&J, + cos@z - xz’)wY
(ga)
and wI= - sin0& + cos&5, -
00se(x+
ZZ’)C+.
@b)
Here S,, 6, are the components of a rigid-body translation and w, is the rigid-body rotation. The rigid-body motion is consistent (i.e. gives zero strains) with the strain-displacement equations (1) but not, of course, with the approximate strain-displacement equations of the preceding section. Reduced rigid-body motions can be obtained from the exact ones by using similar assumptions as were employed in obtaining the reduced strain-displacement equations. Proceeding along these lines leads to the displacement fields l/,=6,+
z’6, +(z-xz’)o,
(94
and 1v,=6z-Xxog
(9b)
for the truly-shallow arch element, together with II,= 6, + z’6, + (z - XZ’)OY
(lOa)
and II‘,= - z’bx + 6, -x0,
(lob)
for the locally-shallow arch. The motions (9) yield vanishing strain and curvature in equations (3) whilst the motions (10) yield finite (but small) values of strain and curvature for the locally-shallow arch (equations (4)). Where an element has constant initial curvature-or, more precisely, constant z”it is possible and more convenient to express the strain-free motions for the truly-shallow and for the locally-shallow arch in another form. For the former arch the motions are
and
564
D. J. DAWE
where a, etc. are constants. For the locally-shallow arch the homogeneous displacement equations are identically satisfied by the motions tl,=a, + (Izsin(z”x) - ascos(z”x)
strain-
(12a)
and w,=azcos(z”x)
+ azsin(z”x).
(12b)
In the alternative formulation where the displacement components are in the Cartesian directions the exact rigid-body motion is
(134 and (13b) and this is unchanged by any shallowness assumptions. This rigid-body motion precisely satisfies the homogeneous strain-displacement equations for the arbitrary arch, the trulyshallow arch and the locally-shallow arch. (For the latter arch this would not be so were the final term of the additional curvature expression of equation (7~) not included; then a very small curvature would result under the rigid-body motion.) 3.2 Basic strain states In this section the displacement patterns corresponding to linearly-varying states of strain and curvature in some of the curved arch elements are presented. These strain patterns are obtained by integrating the assumed strain and curvature variations e=a.++a,x
UW
x=a6 +a+
(14b)
and
where a4 . . . a, are undetermined constants and E and x are defined in terms of the arch displacements and their derivatives as in Section 2. Considering first the arbitrary truly-shallow arch where the arch behaviour is governed by the strain-displacement equations (3), the displacements corresponding to the above strain pattern are
us=
f$x2zr- 1:Ixz’dx)
+ %x3zf-Ji
3x’l’dx)
+a4x+as<
(Isa)
and w,=
a6
a7
-x2 + - x3. 2
6
(15b)
Where the arch has constant initial curvature the tangential displacement can be more conveniently written +a&)
+a,x+asg.
U5c)
Curved Finite Elementsfor the Analysisof Shallowand Deep Arches
565
For the analysis of the arbitrary shallow arch based on Cartesian components of displacement and the strain-displacement equations (6) the displacement components are
and
x2
i3s=a6-2
+a,;.
x3
WI
For the locally-shallow arch of constant initial curvature the displacements corresponding to the assumed strain pattern and the strain-displacement equations (4) are
and
(17b)
ws=
A solution (involving logarithmic and inverse trigonometric functions) can also be found for the locally-shallow arch with Cartesian displacement components and the straindisplacement equations (7). However, this solution has not been incorporated into any finite element model since it will be shown that use of the strain-displacement equations (6) gives adequate results for the deep problems considered.
4. ARCH ELEMENTS In modelling the geometry of the unloaded arch by elements of the Marguerre type the height of the arch, z(x), above the local base line is assumed to be a cubic polynomial whose coefficients are equated to the height and slope at the two end nodes. On the other hand, in using elements of the Vlasov type in the circular arch applications described in Section 5 it is more convenient to input the curvature directly. 4.1 Elements for shallow structures It can be shown that one of the two Euler equations of the shallow curved arch variational problem indicates that the extensional strain is uniform in an interval between loads. This is so whether the arch strain energy is based on the Vlasov strain-displacement equations (3) or the Marguerre equations (6). Thus there is no point in assuming anything higher than a uniform value of extensional strain in a truly-shallow arch element. For the truly-shallow element it has been seen that the displacement components corresponding to strain-free and to uniform-strain, linear-curvature states are of polynomial form whose order depends on the order of the expression for the height z(x). The membrane displacement component corresponding to such states contains higher-order terms than the normal component and this suggests that the usual procedure of adopting a lowerorder independent polynomial expression for the membrane component than for the
D. J. DAWE
566
normal component is not the most efficient one. If the height of the arch centre line above the base line is taken to be a cubic function of x then the independent membrane component of displacement should be of quintic order if it is deemed necessary to accurately represent a state of uniform strain. Eight element models have been developed for truly-shallow arch analysis and with these models the local and global axes coincide. Five of the models (designated Sl to S5) are based on the Vlasov strain-displacement equations (3) whilst the remaining three (S6 to S8) are based on the Marguerre equations (6). The eight elements are as follows:Sl. A ‘quintic-quintic’ element with the independent components of displacement
(184 and (18b) The nodal degrees-of-freedom are U, a’, u”, w, w’ and w” and all are made continuous at the nodes (this implies continuity of rotation, additional curvature and extensional strain in the shallow arch). S2. A ‘quintic-cubic’ element with the components of displacement
+A,X+A3x2+&3
u=A,
(194
and w=Ag+Asx+A,X2fAsX3+AgX4+AIOX5. The nodal degrees-of-freedom the nodes.
(lob)
are U, u’, w, w’, W” and all are made continuous
at
S3. A ‘cubic-cubic’ element with components of displacement
and
The nodal degrees-of-freedom nodes.
are U, u’, w, w’ and are all made continuous at the
S4. A ‘condensed cubic-cubic’ element with components of displacement defined as equation (20) but with the ‘extra’ degree-of-freedom U’ at each node condensed out by minimising the element potential energy, assuming that the potential energy of the loads The final degrees of freedom are thus is independent of the extra degree-of-freedom. only the ones necessary for kinematic compatibility, i.e. U, w, NJ’. S5. A constant-strain, linear-curvature element based on the coupled components of displacement (assembled from equations (11 and 15) with a,=0 in the latter equation),
x2
u=A,,+A,x+z"
A3x+A4y+A5
x3
3+A,g
x4
(214
and w=A3+A4x+A,x2+A,x3.
@lb)
Curved Finite Elements for the Analysis of Shallow and Deep Arches
567
The degrees-of-freedom are again u, w, w’. 5’6. A quintic-quintic element with the Cartesian components of displacement fi and - --I --I, - --I E defined as in equation (18) and with the nodal degrees-of-freedom u, u , II , w, w and ,E”, S7. A cubic-cubic element with the Cartesian components of displacement E and Z - -_I defined as in equation (20) and with the nodal degrees-of-freedom u, u , w and is’. S8. The original constant-strain, linear-curvature element of Ref. [6] with the nodal degrees-of-freedom ii, 9, a’, based on the displacement components (obtained from equations (13 and 16) with a,=O), E=A,+A,x-Ap2A,(
xz-
{;zdx)
-3&(
x’z$Zxrdx)
(224
and iC=A,+A,x+A,x2+A,x3.
(22b)
4.2 Elements for deep structures As with the truly-shallow arch element, models for deep structures are considered which are based on independently-interpolated displacement components or on coupled components describing simple strain states, including the zero-strain state. It is noted that the nature of the strain states is quite different from that associated with the trulyshallow arch (see Section 3) and that the corresponding displacement components are not always of polynomial form. Also, in contrast to the shallow arch, the extensional strain for the deep structure is clearly not uniform, in general, in the interval between loads but this need not mean that an element restricted to a uniform distribution of membrane strain is inefficient. In analysing deep arches the end points of an element lie on the arch centre-line and the local Cartesian axes obviously do not coincide with the global axes. The question arises as to what global degrees-of-freedom should be used. One possibility is to take as degrees-of-freedom the tangential and normal components of displacement together with derivatives of these quantities with respect to the curvilinear coordinate in the arch middle surface. However, connection of such derivatives at the node points means in general that the values of rotation, extensional strain and additional curvature, calculated from the approximate equations given earlier, cannot match. Although, the magnitude of such mis-match will be small (since the individual elements are shallow) it appears more appropriate to directly connect at the nodes quantities such as the (approximate) rotation, strain and curvature as required. The elements considered for the analysis of deep arches are as follows:Dl. A quintioquintic element based on the deep strain-displacement equations (4) (it follows that the assumed components of displacement are the surface ones u and w). The global degrees-of-freedom are u, E, E’, w, 4 and x. 02. A quint&cubic element based on equations (4) and with degrees-of-freedom u, E, w, 4 and x. 03. A cubic-cubic element based on equations (4) and with degrees-of-freedom u, E, w and 4. 04. A condensed cubic-cubic element again based on equations (4) and with degreesof-freedom u, w and 4.
568
D. J. DAWE
D5. Two variations of an assumed-strain model based on equations (4) and the displacement components (from equations (12 and 17)). +A,x +TASX’Z”
u =A, +&sin(z”x)-A,cos(z”xj
W-4
and ~=A~cos(z”~)+A~sin(z”x)+A,
+ A+
W)
In the first approach A, is put equal to A5 to give a constant-strain, linear-curvature element with degrees-of-freedom u, w and 4 at the end nodes. In a second approach AS and A, are independent of one another and the seven coefficients of equation (23) are equated to u, w and 4 at the element end points together with the tangential displacement at the element centre. This latter degree-of-freedom is eliminated by static condensation. 06. A quintic-quintic element with assumed Cartesian components of displacement ii and R. This model is, like element S6, based on the strain-displacement equations (6). The global degrees-of-freedom are U, E, E’, w, 4 and x. 07. A cubic-cubic element with assumed Cartesian components of displacements and with global degrees-of-freedom U, E, w and 4. Variations of this model derived from both shallow and deep expression for the rotation and curvature (equations (6 and 7)) have been considered; these variations are discussed in Section 5. 08. A constant-strain, linear-curvature element as model SS (i.e. based on equations (6) and the assumed displacement components as equation (22)) but with the transformation to global degrees-of-freedom U, w and 4. D9. A constant-strain, linear-curvature element as (shallow) model S5 but used in a deep context with global degrees-of-freedom U, w and 4.
5. NUMERICAL
STUDIES
The models described in the last section have been applied to the solution of a number of arch problems using a standard finite element programme employing double-precision arithmetic throughout. The forces and bending moments are calculated in the usual manner via displacement derivatives. 5.1 Shallow arch The arch is circular with a base length of 34 in. and a central rise of 1.09 in. (corresponding to a radius of 133.1138 in.): it carries a central vertical load and is clamped at its ends. Two arch thicknesses of l/16 in. and 1 in. are considered. In the finite element analysis a symmetric half of the arch is discretised into elements of equal base length whose local axes coincide with the arch global axes. For the thinner arch Fig. 2 illustrates the convergence with mesh refinement of the deflection, w,, under the central load (as a percentage of the exact value) for all the trulyshallow elements. It is noted that due to the approximations inherent in the different shallow strain-displacement equations, the results obtained using Marguerre-type elements (S6-S8) coverge to slightly different values than do the Vlasov type elements (Sl-S5) and that both types of converged solutiondiffer to a small extent from the exact solution. Figure 2 shows that the two quintic-quintic elements, $1 and S6 are very efficient in this application.
Curved Finite Elements for the Analysis of Shallow and Deep Arches
569
When using model Sl the same central deflection, to 6 significant figures, is obtained for all element assemblages from one to 16 elements in the half-arch. When using model S6 there is a small error (0.21 per cent) in the single-element solution but convergence is very rapid; the error is a positive one because of the geometric approximations involved in modelling the true arch. Apart from these elements the two uniform-strain linearcurvature elements converge most rapidly and have a very similar performance. It should be noted that, although differing on a degree-of-freedom basis the results for the Vlasov cubic-cubic element (S3) and the reduced cubic-cubic element (S4) are very similar for the same number of elements. 100 -
1
I
Y
0
I 20
I IO FINAL
FIG. 2. Convergence
DEGREES
I 30 OF FREEDOM.
0
of central deflection for shallow thii arch.
For the thicker arch convergence of the central deffection is very rapid for all element types and there is so little difference between the various models that graphical presentation of the results is superfluous. It is noted that the relative accuracy demonstrated by the variousmodelsin the thin arch analysis is unchanged and that with all models the solutions obtained with only two elements in the half-arch are within 0.35 per cent of the corresponding converged solution. In particular, the single-element results for both quinticquintic models are effectively identical to the converged solutions. Having obtained a broad picture of the convergence characteristics of the various shallow elements using the central deflection as the measure of comparison it is instructive to examine the distribution throughout the arch of some relevant quantities. Fortunately, in view of the number of element types tested, it will not be necessary to present complete results corresponding to all models in order to demonstrate the salient features of the various models. So far as the distribution of displacement components is concerned it is noted that the normal deflection at the centre is a valid indicator of the accuracy of the overall distribution of the components and that these distributions are smooth with no apparent evidence of any significant oscillation.
570
D. J. DAWE
a.DISPLACEHENT
b. BENDING
COMPONENTS.
MOMENT, 1
c. FORCE. !! I
FIG. 3. Distributions using model S3 for shallow thin arch.
In Figs. 3-6 are shown the distributions of the displacement components, bending moment and force obtained using, in turn, the cubic-cubic model S3 and the quintic-cubic model S2 in the analysis of the thin and thick arch. In these (and subsequent) figures AB is the zero line, A being a clamped boundary and B the arch centre line. Note that the two displacement components are not drawn to the same scale, w being very much larger than u (in Figs. 3, 4, 5, 6) the scale of w is 50 (25) times that of u). It can be seen that the displacement distribution of the thin arch is less simple than that of the thick arch and this, in some part, contributes to the reduced accuracy of the results for the thin arch. A prominent feature of these results is the waviness of the force distributions. Severe oscillations of the force occur in analysing the thin arch with few elements. The magnitude of these oscillations reduces with increase in the number of elements and (markedly) with increase in the thickness of the arch. Some waviness is also apparent in the bending moment distribution obtained using the quint&cubic element. Similar oscillatory behaviour of finite element force and moment distributions has been noted by Cowper et al. [2] using a shallow shell model which is broadly a two-dimensional equivalent of the quintic-cubic arch model.
Curved Finite Elements for the Analysis of Shallow and Deep Arches
-
EXACT
---.
n-2
____
571
n=,
a. bISPLACEHENT COMPONENTS
t,.BENOlNG MOMENT,
--____c--c.
FIG. 1.
Distributions
FORCE.
using model S3 for shallow thick arch.
The distributions shown in Figs. 3 and 4 for the Vlasov-type cubic-cubic element S3 are quite similar to those obtained using the reduced cubic-cubic element S4 and the Marguerre-type, cubic-cubic element S7. For all these elements the bending moment distribution is formed of linear segments. Also, the oscillations in the force distribution are common though it is noted that the strain is highly discontinuous at the node points when using model S4 and that the waviness is more marked in using model S7 than in using model S3. The distributions of displacement, moment and force obtained using the quintic-quintic models for both the thin and thick arches are very accurate. With the Vlasov-type element, Sl, the results are virtually indistinguishable from the exact (deep) ones even when using only a single element in the half-arch. The oscillatory behaviour of the force distributions of the other independently-interpolated displacement models is effectively absent (some slight variation of the force is present but at most the waviness is of the order of one part in a million). With the Marguerre-type element, S6, the displacement and moment distribution are, again, virtually indistinguishable from the exact distributions but there is some small, though not insignificant, fluctuation in the force distribution. For the thin arch the range of these fluctuations is f 3.5 per cent using a single element but this reduces rapidly to less than + 0.1 per cent for three elements. For the thick arch the greatest fluctuation, using a single element, is less than 0.07 per cent.
572
D. J. DAWE
-
b.
BENDIWG
EXACT
MOMENT. I
FIG. 5. Distributions using model S? for shallow thin arch.
Finally, the uniform-strain models, S5 and 58, yield moment distributions which are very similar one to another and which differ little from the distributions shown in Figs. 3 and 4 for the Vlasov cubic--cubic element. Highly accurate force distributions are obtained using models S5 and S8; when two or more elements are used in the half-arch the (uniform) magnitude of the force is within half a per cent of the exact value for both the thick and thin arch. 5.2 Deep circular arches Geometrical details of these arches are as those given for the preceding shallow arch except that now the central rise is 17 in. (i.e. the arch is semi-circular). The arch again carries a central vertical load but, as well as the clamped boundary condition, a condition that the ends of the arch are free to slide horizontally in separately applied (this corresponds to the pinched ring). These problems are basically of an inextensional nature. Equal length elements are used in the analysis of the symmetric half-arch and models Dl-D9 are tested in these applications.
Curved Finite Elements for the Analysis of Shallow and Deep Arches
---.-
EXACT nm I
n-t
A:B a. DISPiACEHENl
b. BENDING
COMPONENTS.
MOMENT,
. -___-I-
c.
FORCE
FIG. 6. Distributions
using model S2 for shallow thick arch.
FtG. 7. Convergence of central deflection, thin clamped arch.
573
574
D. J. DAWE
The convergence of the central deflection with the number of elements in the half-arch, n, is shown in Figs. 7-10 for the clamped and the pinched arch, and for the two arch thicknesses (again & in. and 1 in., corresponding to R/t ratios of 272 and 17). With one exception all models shown ultimate convergence to effectively the same deflection in each particular case. As expected, model D9 which is based on shallow expressions of the Vlasov type for the rotation and additional curvature converges not to the true solution but to a level substantially below this. On the other hand the Marguerre-type models 06, 07 and D8 which are based on shallow rotation and curvature expressions do demonstrate convergence to the true, deep solution. As with the use of flat elements the reason for this is presumably due to the coupling effect al the nodes when the local Cartesian displacement components are linked. The Marguerre cubic-cubic model has in fact been tested in these problems with the curvature expression given in turn by equation (6c), by the full equation (7~) and by equation (7~) with the last term neglected. The differences between the three sets of results thus obtained are small and reduce with mesh refinement. The deflection results (leaving model D9 out of further consideration) fall broadly into two groups. Both types of quinticquintic element (Dl and 06) and of assumed-strain element (D5 and 08) yield consistently accurate results, whatever the arch thickness. The other lower-order, independently-interpolated models (02, 03, 04 and 07) are less successful and their accuracy is dependent on the thickness of the arch, being poorer for the thin arch. Of course the use of the approximate strain-displacement equations affects the accuracy of all models, particularly when few elements are used, and probably accounts for most of the error shown for models D1, D5, 06 and 08.
0
I 2
I
I
,
I
I
I
4
6n
a
IO
I2
FIG. 8. Convergence
of central deflection, thick clamped arch.
The quintic-quintic model results for a particular type of analysis are more accurate, on an element basis, than corresponding assumed-strain model results, though only marginally so for the pinched arch; it will be remembered, however, that approximately twice as many degrees-of-freedom are involved in the solutions using the quintic-quintic models. In the applications considered here the convergence of the quintic-quintic and of the
Curved Finite Elements for the Analysis of Shallow and Deep Arches
575
assumed-strain models based on the (shallow) Marguerre-type analysis is more rapid than that of the corresponding models based on the (deep) Vlasov-type analysis. With regard to the Vlasov-type, assumed-strain model D5 it is noted that the results obtained from the assumption of a uniform variation of extensional strain are virtually identical in these particular applications to those obtained from the assumption of a linear variation of strain. The quintiwubic element 02 offers no significant improvement, in these applications, over the corresponding (Vlasov) cubic-cubic elements, 03 and 04. It can be seen that the latter elements yield similar results one to another; relaxation of the continuity of the ‘extra’ degree-of-freedom, E, at the nodes makes little difference to the results. This is the expected result in applications where the loading is such that the strain is, in fact, everywhere continuous. Nevertheless this finding for elements based on the assumption of local shallowness is in direct contrast to that of Ashwell, Sabir and Roberts [7] in their work on circular arches where it is concluded that it is essential to relax the excess continuity condition in deep arch prob1ems.t
0
2
4
6
n
8
IO
I2
16
Fro. 9. Convergence of central deflection, thin pinched arch.
Before leaving the discussion of the displacements it is further noted that the wellknown simple flat element (cubic-linear) has also been used, for comparison purposes, in the analysis of the deep arches. In these applications the performance of the flat element is roughly similar to that of model D5 and is thus superior to all arch models other than those of the assumed-strain or quintic-quintic type.
t These authors have since shown that their conclusion is incorrect in a note (IN. J. Mech. Sci. 15, 325-327, 1973) appearing since the present paper was written.
576
D.
I 0
,,
2
4
FIG. 10. Convergence
o.HODEL
E. HODEL
J. DAWE
I
I
6n
’
I
I IO
I2
16
of central deflection, thick pinched arch.
D5
Dt
b.
HODEL
d. MODEL
DJ
DI
t. HODEL FIG.
11. Moment distributions,
thin clamped arch.
D6
-.-.
-
EXACT n - 2
-_---
n - 4 n - 6
Curved Finite Elements for the Analysis of Shallow and Deep Arches
577
The distribution of relevant quantities throughout a representative deep arch-the clamped one-will now be considered. Again, in this deep problem, the accuracy of the calculated central deflection is a good indicator of the accuracy of the overall distribution of the displacement components. The true distribution of the displacements, for both the thin and thick arches, is roughly of the same form as that illustrated for the shallow, thin arch in Fig. 3a. The bending moment and force distributions calculated using models Dl, 02, 03, D5 and 06 for the two deep, clamped arches are shown in Figs. 11-14. The results for the Marguerre uniform-strain element, 08,are very similar to those shown for model D5 (as indeed are the distributions based on the use of flat elements). The results shown as model D5 are for that version where a uniform strain distribution is assumed ; the linear-strain version yields effectively identical moment distributions and centreelement force values. Model 04 results are closely comparable to the illustrated distributions for model 03 except that the force is not single-valued at the nodes. The nature of the Marguerre cubic-cubic model 07 results is also similar to that of model 03.
-_-._
EXACT n2
----.._ ___.
n=, “-8 h=,6
a.MODEL
D5
c. MODEL
DZ
d.
MODEL
DI
e.
MODEL
D6
*lB
FIG. 12. Force distributions,
thin clamped arch.
The illustrated distributions of moment and force show again that the accuracy of the results obtained using the quintic-quintic models and the assumed-strain (and flat) models is good and is hardly affected by the relative thickness of the arch. Use of the quintic-quintic models yields a very accurate bending moment distribution. The dependence of the accuracy of the quintiocubic and cubic-cubic model results upon the relative thickness of the arch, noted earlier in considering the convergence of the central deflection, is accentuated in these moment and force distributions. For the thin arch the moment distribution calculated using these elements is poor and the fluctuations in the force distributions, using a practical number of elements, are many times greater than the true magnitude of the force. It is recalled, though, that since the problem is primarily a bending one the force is of secondary importance to the bending moment.
D. J. DAWE
a.HODEL
D5
b. MODEL
D3
c. MODEL
DZ
d. MODEL
DI
e. MODEL
FIG. 13. Moment distributions,
a. MODEL
05
D6
thick clamped arch.
b. MODEL
03
d. MODEL
DI
/
EXACT ---n.* --n-4 ____nrb
c. MODEL
D2
FIG. 14. Force distributions,
o. MODEL thick clamped arch.
06
--n-4 ____nx8
Curved Finite Elements for the Analysis of Shallow and Deep Arches
579
6. CONCLUSIONS The numerical results of a representative shallow-arch application show that quinticquintic models, based on either a Marguerre or a Vlasov type of analysis, give excellent results, for thick or thin geometry, even when only a single element is used. The success of these models was anticipated following a consideration of the governing equations of the problem and of the nature of the strain-free and simple straining states of the shallow curved beam element. The assumed-strain models considered also give reliable results though these are not as precise as the quinticquintic model results. The lower-order, independently-interpolated models (quintic-cubic and cubic-cubic) are characterised by the fact that their efficiency is dependent upon the relative thickness of the arch. Calculated force distributions based on these models show oscillatory behaviour which is very marked for a thin arch. No significant improvement in accuracy is achieved by increasing the order of the normal deflection component from cubic to quintic if the tangential component is not similarly increased. In fact it would probably be more efficient to increase the order of the tangential component rather than the normal one (in a cubic-quintic model, say) but this has not been pursued in the present study. The results obtained for the deep-arch applications using locally-shallow elements further emphasise the above findings. The models which successfully deal with both thick and thin deep arches are again the quint&quintic and assumed-strain models. The other models considered (quintic-cubic and cubic-cubic) give force and moment distributions of a very oscillatory nature for the thin arch and it follows that their shell equivalents would exhibit similar tendencies in like circumstances. In considering the shell problem the concept of the assumed-strain model is an attractive one since it requires the use of only a minimal number of degrees-of-freedom. Unfortunately the derivation of such models in a shell analysis of any generality is difficult and the displacement components corresponding to the assumed-strain pattern will, in general, not be compatible at element boundaries. On the other hand the application of independentlyinterpolated displacement fields in shell analysis is more straightforward in concept and reduces compatibility difficulties. The results presented here suggest that for the development of an efficient general shell element all three displacement components should be of essentially quintic order. Such a conclusion might possibly be unduly restrictive, however, since the present work has omitted consideration of any models based on displacement components of quartic order or, as mentioned earlier, of models having a tangential component of higher order than the normal one. The strain-displacement equations which have been used in the present study are approximate ones involving the assumption of shallowness with respect to a local base line. This source of error could be removed, at the expense of increased complexity, by using unreduced strain-displacement equations in a similar analysis and this might be particularly appropriate in shell work. It would not be expected that the conclusions regarding the relative merit of the various types of model would be changed. Consideration of the approximate equations has confirmed that the use of locally-shallow elements based on the standard (shallow) Vlasov strain-displacement equations leads to incorrect solution of deep problems. On the other hand, due to the coupling effect at the nodes on connecting adjacent elements, convergence to the true solution of deep problems can be obtained when using models based on the Marguerre shallow equations.
580 Acknowledgement-This
D. J. DAWE paper is published
by permission
of the Central Electricity Generating
Board.
REFERENCES [I] D. J. DAWE, Curved finite elements in the analysis of shell structures by the finite element method. Pm. 1st Znt. Conf. on Structural Mechanics in Reactor Technology, Vol. 5. Part J, Springer, Berlin (1971). [21 G. R. COWPER,G. M. LINDBERGand M. D. OLSON, A shallow shell finite element of triangular shape. /nt. J. Solids Struct. 6, 1133-l 156 (1970). Pi G. R. COWPER, G. M. LINDBERGand M. D. ORSON, Comparison of two high-precision triangular finite elements for arbitrary deep shells. Proc. 3rd Conf. on Matrix Methods in Structural Mechanics, Wright-Patterson Air For& Ba& Ohio (1971). t41J. H. ARGYRI~and D. W. SCHARPF,The SHEBA family of shell elements for the matrix displacement method. Aero. J. 72, 873-883 (1968). [51G. DUPUI~ and J. J. GOAL, A curved finite element for thin elastic shells. Znt. J. Solids Struct. 6, 1413-1428 (1970). El D. J. DARE, A finite deflection analysis of shallow arches by the discrete element method. Royal Aircraft Establishment (Famborough) Tech. Rep. 70058, April (1970); also published in Int. J. Num. Meth. Engqg 3,529-552 (1971). 171D. G. ASHWELL,A. B. SABCRand T. M. ROBERT&Further studies in the application of curved finite elements to circular arches. Int. J. Mech. Sci. 13,507-517 (1971). 181W. T. KOITER,A consistent first approximation in the general theory of thin elastic shells, Delft (1959). (North-Holland, 1960.) 1V. Z. VLASOV,General theory of shells and its application in engineering. NASA ‘ITF-99 (1964). _[9J [lo] K. MARGUERRE,Zur theorie der gekriimmten Platte grosser formtinderung. Proc. Znt. Co& Appl. Mech., 93-101 (1938). [ll] D. J. DAWE, Rigid body motions and strain-displacement equations of curved shell finite elements. Znt. J. Mech. Sci. 14, 569-578 (1972). (Received 6 March 1973)