Finite Element Method - Shells a special case of three - dimensional analysis - reissner - mindlin assumptions _08

Finite Element Method - Shells a special case of three - dimensional analysis - reissner - mindlin assumptions _08 This monograph presents in detail the novel "wave" approach to finite element modeling of transient processes in solids. Strong discontinuities of stress, deformation, and velocity wave fronts as well as a finite magnitude of wave propagation speed over elements are considered. These phenomena, such as explosions, shocks, and seismic waves, involve problems with a time scale near the wave propagation time. Software packages for 1D and 2D problems yield significantly better results than classical FEA, so some FORTRAN programs with the necessary comments are given in the appendix. The book is written for researchers, lecturers, and advanced students interested in problems of numerical modeling of non-stationary dynamic processes in deformable bodies and continua, and also for engineers and researchers involved designing machines and structures, in which shock, vibro-impact, and other unsteady dynamics and waves processes play a significant role.

Shells as a special case of

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

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}to:~ { !;} bottom )

(8-1)

Here N j ( < , 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

Shell element with displacement and rotation parameters 269

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

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

270 Shells as a special case

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 through-

out 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 direc-

tions normal to the nodal vector V 3 i 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 + y j + z k

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 V l i For example if the x component of V3i is the

smallest one we construct

(8.6)

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

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 assump-

tions

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)

272 Shells as a special case

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

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 is only approximately normal to the mid-surface

Shell element with displacement and rotation parameters 273 where the matrix H is a function of the coordinates For instance, in the stiffness

matrix

and with the usual definition of Chapter 2 of Volume 1,

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 V I 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

274 Shells as a special case

From this the components of the B matrix can now be found explicitly, noting that 5

degrees of freedom exist at each node:

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

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

276 Shells as a special case

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

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