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Shells as a special case of three-dimensional analysis -

Reissner-Mindlin assumptions

8.1 Introduction In the analysis of solids the use of isoparametric, curved two- and three-dimensional elements is particularly effective, as illustrated in Chapters 1 and 3 and presented in Chapters 9 and 10 of Volume 1. It seems obvious that use of such elements in the analysis of curved shells could be made directly simply by reducing their dimension in the thickness direction as shown in Fig. 8.1. Indeed, in an axisymmetric situation such an application is illustrated in the example of Fig. 9.25 of Volume 1. With a straightforward use of the three-dimensional concept, however, certain difficulties will be encountered.

In the first place the retention of 3 displacement degrees of freedom at each node leads to large stiffness coefficients from strains in the shell thickness direction. This presents numerical problems and may lead to ill-conditioned equations when the shell thickness becomes small compared with other dimensions of the element.

The second factor is that of economy. The use of several nodes across the shell thickness ignores the well-known fact that even for thick shells the ‘normals’ to the mid-surface remain practically straight after deformation. Thus an unnecessarily high number of degrees of freedom has to be carried, involving penalties of computer time.

In this chapter we present specialized formulations which overcome both of these difficulties. The constraint of straight ‘normals’ is introduced to improve economy and the strain energy corresponding to the stress perpendicular to the mid-surface is ignored to improve numerical ~onditioning.’-~ With these modifications an efficient tool for analysing curved thick shells becomes available. The accuracy and wide range of applicability of the approach is demonstrated in several examples.

8.2 Shell element with displacement and rotation parameters

The reader will note that the two constraints introduced correspond precisely to the so- called Reissner-Mindlin assumptions already discussed in Chapter 5 to describe the

Shell element with displacement and rotation parameters 267

Fig. 8.1 Curved, isoparametric hexahedra in a direct approximation to a curved shell.

behaviour of thick plates. The omission of the third constraint associated with the thin plate theory (normals remaining normal to the mid-surface after deformation) permits the shell to experience transverse shear deformations - an important feature of thick shell situations.

The formulation presented here leads to additional complications compared with the straightforward use of a three-dimensional element. The elements developed here are in essence an alternative to the processes discussed in Chapter 5, for which an independent interpolation of slopes and displacement are used with a penalty function imposition of the continuity requirements. The use of reduced integration is useful if thin shells are to be dealt with - and, indeed, it was in this context that this procedure was first discovered.4-’ Again the same restrictions for robust behaviour as those discussed in Chapter 5 become applicable and generally elements that perform well in plate situations will do well in shells.

268 Shells as a special case

8.2.1 Geometric definition of an element

Consider a typical shell element illustrated in Fig. 8.2. The external faces of the element are curved, while the sections across the thickness are generated by straight lines. Pairs of points, itop and ibottom, each with given Cartesian coordinates, prescribe the shape of the element.

Let <, r] be the two curvilinear coordinates in the mid-surface of the shell and let C be a linear coordinate in the thickness direction. If, further, we assume that <, r] , C vary between -1 and 1 on the respective faces of the element we can write a relationship between the Cartesian coordinates of any point of the shell and the curvilinear coordinates in the form

{ :} = c N i ( < i V ) (7 { ! i } t o : ~ { !;} ) bottom

(8-1)

Here Nj(< , r ] ) is a standard two-dimensional shape function taking a value of unity at the top and bottom nodes i and zero at all other nodes (Chapter 9 of Volume 1). If the basic functions Ni are derived as ‘shape functions’ of a ‘parent’, two-dimensional

Fig. 8.2 Curved thick shell elements of various types.


Fig. 8.3 Local and global coordinates.

element, square or triangular: in plan, and are so ‘designed’ that compatibility is achieved at interfaces, then the curved space elements will fit into each other. Arbitrary curved shapes of the element can be achieved by using shape functions of higher order than linear. Indeed, any of the two-dimensional shape functions of Chapter 8 of Volume 1 can be used here.

The relation between the Cartesian and curvilinear coordinates is now established and it will be found desirable to operate with the curvilinear coordinates as the basis. It should be noted that often the coordinate direction c is only approximately normal to the mid-surface.

It is convenient to rewrite the relationship, Eq. (8.1), in a form specified by the ‘vector’ connecting the upper and lower points (i.e. a vector of length equal to the shell thickness t ) and the mid-surface coordinates. Thus we can rewrite Eq. (8.1) as (Fig. 8.3)+

{ !} = c Ni(<>q) ({ !i}mi:+ (V3i) (8.2)

where

{ !:} = + ({ _1) +{a) ) and v3i= { ::} -{ !:} top bottom top bottom

(8.3)

with V3i defining a vector whose length represents the shell thickness.

* Area coordinates L, would be used in this case in place of I, 7 as in Chapter 8 of Volume I . t For details of vector algebra see Appendix F of Volume I .


For relatively thin shells, it is convenient to replace the vector V3i by a unit vector v3i in the direction normal to the mid-surface. Now Eq. (8.2) is written simply as

where ti is the shell thickness at the node i. Construction of a vector normal to the mid-surface is a simple process (see Sec. 6.4.2).

8.2.2 Displacement field

The displacement field is now specified for the element. As the strains in the direction normal to the mid-surface will be assumed to be negligible, the displacement throughout the element will be taken to be uniquely defined by the three Cartesian components of the mid-surface node displacement and two rotations about two orthogonal directions normal to the nodal vector V3i . If these two orthogonal directions are denoted by unit vectors v l i and v2i with corresponding rotations ai and pi (see Fig. 8.3), we can write, similar to Eq. (8.2) but dropping the subscript ‘mid’ for simplicity,

from which the usual form is readily obtained as

where u, w and w are displacements in the directions of the global x, y and z axes. As an infinity of vector directions normal to a given direction can be generated, a

particular scheme has to be devised to ensure a unique definition. Some such schemes were discussed in Chapter 6. Here another unique alternative will be given,234 but other possibilities are open.7

Here V3i is the vector to which a normal direction is to be constructed. A coordinate vector in a Cartesian system may be defined by

x = xi + yj + zk

in which i, j and k are three (orthogonal) base vectors. To find the first normal vector we find the minimum component of V3i and construct a vector cross-product with the unit vector in this direction to define Vli . For example if the x component of V3i is the smallest one we construct

(8.6)

Vl i = i x V3i (8.7)

Shell element with displacement and rotation parameters 27 1

where

i = [ ~ o 0IT

is the form of the unit vector in the x direction. Now

defines the first unit vector. The second normal vector may now be computed from

vzi = v 3 i x Vli (8.9)

and normalized using the form in Eq. (8.8). We have thus three local, orthogonal axes defined by unit vectors

v l i , v2i and v3i (8.10)

Once again if Ni are C, functions then displacement compatibility is maintained between adjacent elements.

The element coordinate definition is now given by the relation Eq. (8.2) and has more degrees of freedom than the definition of the displacements. The element is therefore of the ‘superparametric’ kind (see Chapter 9 of Volume 1) and the constant strain criteria are not automatically satisfied. Nevertheless, it will be seen from the definition of strain components involved that both rigid body motions and constant strain conditions are available.

Physically it has been assumed in the definition of Eq. (8.4) that no strains occur in the ‘thickness’ direction C. While this direction is not always exactly normal to the mid-surface it still represents a good approximation of one of the usual shell assumptions.

At each mid-surface node i of Fig 8.3 we now have the 5 basic degrees-of-freedom, and the connection of elements will follow precisely the patterns described in Chapter 6 (Secs 6.3 and 6.4).

8.2.3 Definition of strains and stresses

To derive the properties of a finite element the essential strains and stresses need first to be defined. The components in directions of orthogonal axes related to the surface < (constant) are essential if account is to be taken of the basic shell assumptions. Thus, if at any point in this surface we erect a normal 2 with two other orthogonal axes X and 7 tangential to it (Fig. 8.3), the strain components of interest are given simply by the three-dimensional relationships in Chapter 6 of Volume 1:

(8.11)


- D = - E

1 - v*

with the strain in direction Z neglected so as to be consistent with the usual shell assumptions. It must be noted that in general none of these directions coincide with those of the curvilinear coordinates E, q, <, although X, J are in the Eq plane (< = constant).*

The stresses corresponding to these strains are defined by a matrix 0 and for elastic behaviour are related to the usual elasticity matrix D. Thus

- ‘1 v 0 0 0

v 1 0 0 0 0 0 (1-v) /2 0 0 (8.13) 0 0 0 K ( l - v ) / 2 0

- 0 0 0 0 K ( 1 - v) /2 -

8.2.4 Element properties and necessary transformations

The stiffness matrix - and indeed all other ‘element’ property matrices - involve integrals over the volume of the element, which are quite generally of the form

(8.14)

* Indeed, these directions will only approximately agree with the nodal directions v,,. v2, previously derived, as in general the vector vJi is only approximately normal to the mid-surface.


where the matrix H is a function of the coordinates. For instance, in the stiffness matrix

H = BTDB (8.15)

and with the usual definition of Chapter 2 of Volume 1, T. = Ba' (8.16)

we have B defined in terms of the displacement derivatives with respect to the local Cartesian coordinates X, j , Z by Eq. (8.11). Now, therefore, two sets of transformations are necessary before the element can be integrated with respect to the curvilinear coordinates 5, v, (.

First, by identically the same process as we used in Chapter 9 of Volume 1, the derivatives with respect to the x, y, z directions are obtained. As Eq. (8.4) relates the global displacements u, w, w to the curvilinear coordinates, the derivatives of these displacements with respect to the global x, y , z coordinates are given by a matrix relation:

In this, the Jacobian matrix is defined as

(8.17)

(8.18)

and calculated from the coordinate definitions of Eq. (8.2). Now, for every set of curvilinear coordinates the global displacement derivatives can be obtained numerically.

A second transformation to the local displacements X, j , Z will allow the strains, and hence the B matrix, to be evaluated. The directions of the local axes can be established from a vector normal to the 57 mid-surface (( = 0). This vector can be found from two vectors xx and x , ~ that are tangential to the mid-surface. Thus

v3 = [ ;;] x [ ;;] = [ zxx,o -z,qx><]

We can now construct two perpendicular vectors VI and V2 following the process given previously to describe the x and j directions, respectively. The three orthogonal vectors can be reduced to unit magnitudes to obtain a matrix of vectors in the X, J , Z directions (which is in fact the direction cosine matrix) given as

@ = [ V I , v 2 , v31 (8.20)

The global derivatives of displacement u, TJ and w are now transformed to the local

yxz>a - YJlZX

XLY, - X>?7Y><

(8.19)

derivatives of the local orthogonal displacements by a standard operation


From this the components of the B matrix can now be found explicitly, noting that 5 degrees of freedom exist at each node:

E = Ba‘ (8.22)

where the form of ae is given in Eq. (8.5). The infinitesimal volume is given in terms of the curvilinear coordinates as

dxdydz = det (JId[dqd< =jd[dqd[ (8.23)

where j = det I JI. This standard expression completes the basic formulation. Numerical integration within the appropriate limits is carried out in exactly the

same way as for three-dimensional elements using the Gaussian quadrature formulae discussed in Chapter 9 of Volume 1. An identical process serves to define all the other relevant element matrices arising from body and surface loading, inertia matrices, etc.

As the variation of the strain quantities in the thickness, or [direction, is linear, two Gauss points in that direction are sufficient for homogeneous elastic sections, while three or four in the [, q directions are needed for parabolic and cubic shape functions N j , respectively.

It should be remarked here that, in fact, the integration with respect to < can be performed explicitly if desired, thus saving computation time. 134

8.2.5 Some remarks on stress representation

The element properties are now defined, and the assembly and solution are in standard form. It remains to discuss the presentation of the stresses, and this problem is of some consequence. The strains being defined in local direction, a, are readily available. Such components are indeed directly of interest but as the directions of local axes are not easily visualized (and indeed may not be continuously defined between adjacent elements) it is sometimes convenient to transfer the components to the global system using the standard transformation

(8.24)

Such a transformation should be performed only for elements which belong to the approximation for the same smooth surface.

In a general shell structure, the stresses in a global system do not, however, give a clear picture of shell surface stresses. It is thus convenient always to compute the principal stresses (or invariants of stress) by a suitable transformation. Regarding the shell stresses more rationally, one may note that the shear components T~~ and rjz are in fact zero on the top and bottom surfaces and this may be noted when making the transformation of Eq. (8.24) before converting to global components to ensure that the principal stresses lie on the surface of the shell. The values obtained directly for these shear components are the average values across the section. The maximum transverse shear on a solid cross-section occurs on the mid-surface and is equal to about 1.5 times the average value.

Special case of axisymmetric, curved, thick shells 275

8.3 Special case of axisymmelic, curved, thick sheHs For axisymmetric shells the formulation is simplified. Now the element mid-surface is defined by only two coordinates <, r] and a considerable saving in computer effort is obtained.'

The element now is derived in a similar manner by starting from a two-dimensional definition of Fig. 8.4.

Equations (8.1) and (8.2) are now replaced by their two-dimensional equivalents defining the relation between coordinates as

{ I} = C Ni(<) ( 7 { :} +? { :} bottom ) top

(8.25) ) = C Ni(<) ({ :} +t V t i V g i mid

Fig. 8.4 Coordinates for an axisymmetric shell: (a) coordinate representation; (b) shell representation.


with cos q5i

sin q5i v3i = { } in which $i is the angle defined in Fig. 8.4(b) and ti is the shell thickness. Similarly, the displacement definition is specified by following the lines of Eq. (8.4).

Here we consider the case of axisymmetric loading only. Non-axisymmetric loading is addressed in Chapter 9 along with other schemes which permit treatment of problems in a reduced manner. Thus, we specify the two displacement components as

{3 = C N i ( { :}+${ cos 4i } p i ) (8.26)

In this pi stands for the rotation illustrated in Fig. 8.5, and ui, wi stand for the displacement of the middle surface node.

- sin 4i

Global strains are conveniently defined by the relationship'

.={;;)-( 5 ] (8.27)

These strains are transformed to the local coordinates and the component normal to q (q = constant) is neglected.

All the transformations follow the pattern described in previous sections and need not be further commented on except perhaps to remark that they are now carried out only between sets of directions <,q, r,z, and T,F, thus involving only two variables.

Similarly the integration of element properties is carried out numerically with respect to < and q only, noting, however, that the volume element is

(8.28)

Yrz u,z + w,r

dx dy dz = det I JI d< dqr dB = j r d< dq de

Fig. 8.5 Global displacements in an axisymmetric shell.

Special case of thick plates 277

Fig. 8.6 Axisymmetric shell elements: (a) linear; (b) parabolic; (c) cubic.

By suitable choice of shape functions N j ( < ) , straight, parabolic, or cubic shapes of variable thickness elements can be used as shown in Fig. 8.6.

8.4 Special case of thick plates The transformations necessary in this chapter are somewhat involved and the programming steps are quite sophisticated. However, the application of the principle involved is available for thick plates and readers are advised to first test their com- prehension on such a simple problem.

1. [ = 2 z / t and unit vectors v l , v2 and v3 can be taken in the directions of the x, y , and

2. ai and pi are simply the rotations 0, and Ox, respectively (see Chapter 5). 3. It is no longer necessary to transform stress and strain components to a local

system of axes 2, J , Z and global definitions x, y , z can be used throughout. For elements of this type, numerical thickness integration can be avoided and, as an exercise, readers are encouraged to derive the stiffness matrices, etc., for, say, linear, rectangular elements. Forms will be found which are identical to those derived in Chapter 5 with an independent displacement and rotation interpolation

Here the following obvious simplifications arise.

z axes respectively.


and using shear constraints. This demonstrates the essential identity of the alternative procedures.

8.5 Convergence Whereas in three-dimensional analysis it is possible to talk about absolute convergence to the true exact solution of the elasticity problem, in equivalent plate and shell problems such a convergence cannot happen. As the element size decreases the so- called convergent solution of a plate bending problem approaches only to the exact solution of the approximate model implied in the formulation. Thus, here again convergence of the above formulation will only occur to the exact solution constrained by the requirement that straight ‘normals’ remain straight during deformation.

In elements of finite size it will be found that pure bending deformation modes are nearly always accompanied by some shear strains which in fact do not exist in the conventional thin plate or shell bending theory (although quite generally shear stresses may be deduced by equilibrium considerations on an element of the model, similar to the manner by which shear stresses in beams are deduced). Thus large elements deform- ing mainly under bending action (as would be the case of the shell element degenerated to a flat plate) tend to be appreciably too stiff. In such cases certain limits of the ratio of size of element to its thickness need to be imposed. However, it will be found that such restrictions often are relaxed by the simple expedient of reducing the integration order.4

Figure 8.7 shows, for instance, the application of the quadratic eight-node element to a square plate situation. Here results for integration with 3 x 3 and 2 x 2 Gauss points

Fig. 8.7 A simply supported square plate under uniform load qo: plot of central deflection w, for eight-node elements with (a) 3 x 3 Gauss point integration and (b) with 2 x 2 (reduced) Gauss point integration. Central deflection is wc for thin plate theory.

Inelastic behaviour 279

are given and results plotted for different thickness-to-span ratios. For reasonably thick situations, the results are similar and both give the additional shear deformation not available by thin plate theory. However, for thin plates the results with the more exact integration tend to diverge rapidly from the now correct thin plate results whereas the reduced integration still gives excellent results. The reasons for this improved performance are fully discussed in Chapter 2 and the reader is referred there for further plate examples using different types of shape functions.

8.6 Inelastic behaviour All the formulations presented in this chapter can of course be used for all non-linear materials. The procedures are similar to those mentioned in Chapters 4 and 5 dealing with plates. Now it is only necessary to replace Eqs (8.12) and (8.13) by the appropriate constitutive equation and tangent operator, respectively. In this case it is necessary always to perform the through-thickness integration numerically since a priori knowl- edge of the behaviour will not be available. Any of the constitutive models described in Chapter 3 may be used for this purpose provided appropriate transformations are made to make oz zero.

Fig. 8.8 Spherical dome under uniform pressure analysed with 24 cubic elements (first element subtends an angle of 0.1" from fixed end, others in arithmetic progression).

8.7 Some shell examples A limited number of examples which show the accuracy and range of application of the axisymmetric shell formulation presented in this chapter will be given. For a fuller selection the reader is referred to references 1-7.

Fig. 8.9 Thin cylinder under a unit radial edge load.

Some shell examples 281

8.7.1 Spherical dome under uniform pressure

The ‘exact’ solution of shell theory is known for this axisymmetric problem, illustrated in Fig. 8.8. Twenty-four cubic-type elements are used with graded size more closely spaced towards supports. Contrary to the ‘exact’ shell theory solution, the present formulation can distinguish between the application of pressure on the inner and outer surfaces as shown in the figure.

Fig. 8.10 Cylindrical shell example: self-weight behaviour.


8.7.2 Edge loaded cylinder

A further axisymmetric example is shown in Fig. 8.9 to study the effect of subdivision. Two, six, or fourteen cubic elements of unequal length are used and the results for both of the finer subdivisions are almost coincident with the exact solution. Even the two-element solution gives reasonable results and departs only in the vicinity of the loaded edge.

Once again the solutions are basically identical to those derived with independent slope and displacement interpolation in the manner presented in Chapter 5.

8.7.3 Cylindrical vault

This is a test example of application of the full process to a shell in which bending action is dominant as a result of supports restraining deflection at the ends (see also Sec. 6.8.2).

Fig. 8.1 1 Displacement (parabolic element), cylindrical shell roof.

Some shell examples 283

In Fig. 8.10 the geometry, physical details of the problem, and subdivision are given, and in Fig. 8.11 the comparison of the effects of 3 x 3 and 2 x 2 integration using eight-node quadratic elements is shown on the displacements calculated. Both integrations result, as expected, in convergence. For the more exact integration, this is rather slow, but, with reduced integration order, very accurate results are obtained, even with one element. The improved convergence of displacements is matched by rapid convergence of stress components.

This example illustrates most dramatically the advantages of this simple expedient and is described more fully in references 4 and 6 . The comparison solution for this problem is one derived along more conventional lines by Scordelis and LO.^

8.7.4 Curved dams

All the previous examples were rather thin shells and indeed demonstrated the applicability of the process to these situations. At the other end of the scale, this formulation has been applied to the doubly curved dams illustrated in Chapter 9 of Volume 1 (Fig. 9.28). Indeed, exactly the same subdivision is again used and results reproduce

Fig. 8.12 An analysis of cylinder intersection by means of reduced integration shell-type elernents.’0


almost exactly those of the three-dimensional s o l ~ t i o n . ~ This remarkable result is achieved at a very considerable saving in both degrees of freedom and computer solution time.

Clearly, the range of application of this type of element is very wide.

8.7.5 Pipe penetration" and spherical cap7

The last two examples, shown in Figs 8.12-8.14, illustrate applications in which the irregular shape of elements is used. Both illustrate practical problems of some interest and show that with reduced integration a useful and very general shell element is available, even when the elements are quite distorted.

Fig. 8.13 Cylinder-to-cylinder intersections of Fig. 8.1 2: (a) hoopstresses near 0" line; (b) axial stresses near 0" line.

Concluding remarks 285

Fig. 8.14 A spherical cap analysis with irregular isoparametric shell elements using full 3 x 3 and reduced 2 x 2 integration.

8.8 Concluding remarks The elements described in this chapter using degeneration of solid elements are shown in plate and axisymmetric problems to be nearly identical to those described in Chapters 5 and 7 where an independent slope and displacement interpolation is directly used in the middle plane. For the general curved shell the analogy is less obvious but clearly still exists. We should therefore expect that the conditions established in Chapter 5 for robustness of plate elements to be still valid. Further,


it appears possible that other additional conditions on the various interpolations may have to be imposed in curved element forms. Both statements are true. The eight- and nine-node elements which we have shown in the previous section to perform well will fail under certain circumstances and for this reason many of the more successful plate elements also have been adapted to the shell problem.

The introduction of additional degrees of freedom in the interior of the eight-node serendipity element was first suggested by and later by with- out, however, achieving complete robustness. The full lagrangian cubic interpolation as shown in Chapter 5 is quite effective and has been shown to perform well. How- ever, the best results achieved to date appear to be those in which ‘local constraints’ are applied (see Sec. 5.5) and such elements as those due to Dvorkin and Bathe,16 Huang and Hinton,17 and Simo et ~ l . ” ~ ’ ~ fall into this category.

While the importance of transverse shear strain constraints is now fully under- stood, the constraints introduced by the ‘in-plane’ (membrane) stress resultants are less amenable to analysis (although the elastic parameters Et associated with these are of the same order as those of shear Gt). It is well known that membrane locking can occur in situations that do not permit inextensional bending. Such locking has been thoroughly but to date the problem has not been rigorously solved and further developments are required.

Much effort is continuing to improve the formulation of the processes described in this chapter as they offer an excellent solution to the curved shell p r ~ b l e m . ’ ~ - ~ ~

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