Finite Element Method - Axisymmetric shells _07

Finite Element Method - Axisymmetric shells _07 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.

7

Axi s y m m et r i c s h e I I s

7.1 Introduction

The problem of axisymmetric shells is of sufficient practical importance to include

in this chapter special methods dealing with their solution While the general method described in the previous chapter is obviously applicable here, it will be found that considerable simplification can be achieved if account is taken of axial symmetry of the structure In particular, if both the shell and the loading are axisymmetric it will be found that the elements become ‘one-dimensional’ This is the simplest type of element, to which little attention was given in earlier chapters

The first approach to the finite element solution of axisymmetric shells was presented by Grafton and Strome.’ In this, the elements are simple conical frustra and a direct approach via displacement functions is used Refinements in the derivation of the element stiffness are presented in Popov et a1.* and in Jones and S t r ~ m e ~ An extension to the case of unsymmetrical loads, which was suggested in Grafton and Strome, is elaborated in Percy et d4 and

Later, much work was accomplished to extend the process to curved elements and indeed to refine the approximations involved The literature on the subject is considerable, no doubt promoted by the interest in aerospace structures, and a complete bibliography is here impractical References 7- 15 show how curvilinear coordinates of various kinds can be introduced to the analysis, and references 9 and 14 discuss the use of additional nodeless degrees of freedom in improving accuracy ‘Mixed’ formulations (Chapter 11 of Volume 1) have found here some use.I6 Early work on the subject is reviewed comprehensively by Gallagher’7.18 and Stricklin I 9

In axisymmetric shells, in common with all other shells, both bending and ‘in- plane’ or ‘membrane’ forces will occur These will be specified uniquely in terms

of the generalized ‘strains’, which now involve extensions and changes in curvatures

of the middle surface If the displacement of each point of the middle surface is specified, such ‘strains’ and the internal stress resultants, or simply ‘stresses’, can

be determined by formulae available in standard texts dealing with shell

Straight element 245

7.2 Straight element

As a simple example of an axisymmetric shell subjected to axisymmetric loading we

consider the case shown in Figs 7.1 and 7.2 in which the displacement of a point

on the middle surface of the meridian plane at an angle 4 measured positive from

the x-axis is uniquely determined by two components ii and E in the tangential (s)

and normal directions, respectively

Using the Kirchhoff-Love assumption (which excludes transverse shear deforma-

tions) and assuming that the angle 4 does not vary (i.e elements are straight), the four

strain components are given by2'-**

X S -(dE/ds) COS d/Y

This results in the four internal stress resultants shown in Fig 7.1 that are related to

the strains by an elasticity matrix D:

the upper part being a plane stress and the lower a bending stiffness matrix with shear terms omitted as 'thin' conditions are assumed

Let the shell be divided by nodal circles into a series of conical frustra, as shown in

Fig 7.2 The nodal displacements at points 1 and 2 for a typical 1-2 element such

as i a n d j will have to define uniquely the deformations of the element via prescribed shape functions

At each node the radial and axial displacements, u and w , and a rotation, ,B, will be used as parameters From virtual work by edge forces we find that all three compo- nents are necessary as the shell can carry in-plane forces and bending moments The displacements of a node i can thus be defined by three components, the first two being

in global directions Y and z ,

The simplest elements with two nodes, i and j , thus possess 6 degrees of freedom,

determined by the element displacements

The displacements within the element have to be uniquely determined by the nodal displacements a' and the position s (as shown in Fig 7.2) and maintain slope and displacement continuity

Thus in local (s) coordinates we have

Straight element 247 Based on the strain-displacement relations (7.1) we observe that ii can be of Co type

while I3 must be of type C1 The simplest approximation takes ii varying linearly with

s and ii, as cubic in s We shall then have six undetermined constants which can be

determined from nodal values of u, w , and p

At the node i,

COS$ sin$ 0

(7.7)

Introducing the interpolations

where Ny are the usual linear interpolations in E (-1 d < d 1)

in which, placing the origin of the meridian coordinate s at the i node,

The global coordinates for the conical frustrum may also be expressed by using the

Ny interpolations as

and used to compute the length L as

L = d ( r 2 - r1)2 + (z2 -

Writing the interpolations as

we can now write the global interpolation as

u = { i} = [ N I T N 2 T ] a e = N a e

(7.10)

(7.11)

(7.12)

From Eq (7.12) it is a simple matter to obtain the strain matrix B by use of the

definition Eq (7.1) This gives

-

-

- - - -

with E varying from -1 to 1

Thus, the stiffness matrix K becomes, in local coordinates,

value of the integrand (one-point Gaussian quadrature) and using a D matrix

corresponding to an orthotropic material Percy et d4 and Klein' used a seven- point numerical integration; however, it is generally recommended to use only two- points to obtain all arrays (especially if inertia forces are added, since one point then would yield a rank deficient mass matrix)

It should be remembered that if any external line loads or moments are present, their full circumferential value must be used in the analysis, just as was the case with axisymmetric solids discussed in Chapter 5 of Volume 1

A slight improvement to the above element may be achieved by adding an enhurzceci srrain mode to the E,, component Here this is achieved by following the procedures

Straight element 249

outlined in Chapter 12 of Volume 1, and we can observe that the necessary condition

not to affect a constant value of is given by

(7.18)

where &$en) denotes the enhanced strain component A simple mode may thus be

defined as

(7.19)

in which aen is a parameter to be determined For the linear elastic case considered

above the mode may be determined from

(7.20) where

Now a partial solution may be performed by means of static c ~ n d e n s a t i o n ~ ~ to obtain

the stiffness for assembly

K = K - G T K i ' G (7.22) The effect of the added mode is most apparent in the force resultant N,\ where solu-

tion oscillations are greatly reduced This improvement is not needed for the purely

elastic case but is more effective when the material properties are inelastic where the

oscillations can cause errors in behaviour, such as erratic yielding in elasto-plastic

solutions

7.2.3 Examples and accuracy

In the treatment of axisymmetric shells described here, continuity between the shell

elements is satisfied at all times For an axisymmetric shell of polygonal meridian

shape, therefore, convergence will always occur

The problem of the physical approximation to a curved shell by a polygonal shape

is similar to the one discussed in Chapter 6 Intuitively, convergence can be expected,

and indeed numerous examples indicate this

When the loading is such as to cause predominantly membrane stresses, discrepan- cies in bending moment values exist (even with reasonably fine subdivision) Again, however, these disappear as the size of the subdivisions decreases, particularly if correct sampling is used (see Chapter 14 of Volume 1) This is necessary to eliminate the physical approximation involved in representing the shell as a series of conical frustra

Figures 7.3 and 7.4 illustrate some typical examples taken from the Grafton and

Strome paper which show quite remarkable accuracy In each problem it should be noted that small elements are needed near free edges to capture the ‘boundary layer’ nature of shell solutions

Fig 7.3 A cylindrical shell solution by finite elements, from Grafton and Strorne.’

Curved elements 251

Fig 7.4 A hemispherical shell solution by finite elements, from Grafton and Strome.’

Use of curved elements has already been described in Chapter 9 of Volume 1, in the

context of analyses that involved only first derivatives in the definition of strain Here

second derivatives exist [see Eq (7.1)] and some of the theorems of Chapter 8 of

Volume 1 are no longer applicable

It was previously mentioned that many possible definitions of curved elements have

been proposed and used in the context of axisymmetric shells The derivation used

Fig 7.5 Curved, isoparametric, shell element for axisymmetric problems: (a) parent element; (b) curvilinear coordinates

here is one due to Delpak14 and, to use the nomenclature of Chapter 8, Volume 1, is of the subparametric type

The basis of curved element definition is one that gives a common tangent between adjacent elements (or alternatively, a specified tangent direction) This is physically necessary to avoid ‘kinks’ in the description of a smooth shell

If a general curved form of a shell of revolution is considered, as shown in Fig 7.5, the expressions for strain quoted in Eq (7.1) have to be modified to take into account the curvature of the shell in the meridian plane.20’21 These now become

E = { ii} = { [ii cos diilds q5 - + W W/RS sin q5]/r ]

-d2W/ds2 - d(ii/R,)/ds -[(dii,/ds + ii/R,T)] cos q5/r

The reader can verify that for R, = 00 Eq (7.23) coincides with Eq (7.1)

We shall now consider the 1-2 element to be curved as shown in Fig.7.5(b), where the coordinate is in ‘parent’ form (-1 < < < 1) as shown in Fig 7.5(a) The coordinates

and the unknowns are ‘mapped’ in the manner of Chapter 9 of Volume 1 As we wish

Curved elements 253

to interpolate a quantity with slope continuity we can write for a typical function I/I

(7.24)

where again the order 1 Hermitian interpolations have been used We can now simulta-

neously use these functions to describe variations of the global displacements u and was*

(7.25)

and of the coordinates r and z which define the shell (mid-surface) Indeed, if the

thickness of the element is also variable the same interpolation could be applied to

it Such an element would then be isoparametric (see Chapter 9 of Volume 1) Accord-

ingly, we can define the geometry as

and, provided the nodal values in the above can be specified, a one-to-one relation

between and the position on the curved element surface is defined [Fig 7.5(b)]

While specification of ri and zi is obvious, at the ends only the slope

cotc#)j = - -

is defined The specification to be adopted with regard to the derivatives occurring in

Eq (7.26) depends on the scaling of along the tangent length s Only the ratio

(7.28)

is unambiguously specified Thus (dr/d<)i or (dzldt); can be given an arbitrary value

Here, however, practical considerations intervene as with the wrong choice a very

uneven relationship between s and t will occur Indeed, with an unsuitable choice

the shape of the curve can depart from the smooth one illustrated and loop between

the end values

To achieve a reasonably uniform spacing it suffices for well-behaved surfaces to

* One immediate difference will be observed from that of the previous formulation Now both displacement

components vary in a cubic manner along an element while previously a linear variation of the tangential

displacement was permitted This additional degree of freedom does not, however, introduce excessive constraints provided the shell thickness is itself continuous

using whichever is largest and noting that the whole range of < is 2 between the nodal points

The variation of global displacements are specified by Eq (7.25) while the strains are described in locally directed displacements in Eq (7.23) Some transformations are therefore necessary before the strains can be determined

We can express the locally directed displacements U and W in terms of the global displacements by using Eq (7.7), that is,

cos4 s i n 4 1 { :> = Tu

where 4 is the angle of the tangent to the curve and the r axis (Fig 7.5) We note that

this transformation may be expressed in terms of the < coordinate using Eqs (7.27) and (7.28) and the interpolations for r a n d z With this transformation the continuity

of displacement between adjacent elements is achieved by matching the global nodal displacements ui and wi However, in the development for the conical element we have specified continuity of rotation of the cross-section only Here we shall allow usually the continuity of both s derivatives in displacements Thus, the parameters

in which all the derivatives are directly determined from expression (7.26)

If shells that branch or in which abrupt thickness changes occur are to be treated, the nodal parameters specified in Eq (7.32) are not satisfactory It is better to rewrite these as

ai = [ u i wi Pi (dii/ds)ilT (7.34)

Curved elements 255

where pi, equal to (diV/ds)i, is the nodal rotation, and to connect only the first three

parameters The fourth is now an unconnected element parameter with respect to

which, however, the usual treatment is still carried out Transformations needed in

the above are implied in Eq (7.7)

In the derivation of the B matrix expressions which define the strains, both first and

second derivatives with respect to s occur, as seen in the definition of Eq (7.23) If we

observe that the derivatives can be obtained by the simple (chain) rules already

implied in Eq (7.31), for any function F we can write

(7.35)

d F d F ds d 2 F d 2 F ds d F d2s d[ ds d[

the integral expressions prohibit exact integration, and numerical quadrature must be

used As this is carried out in one coordinate only it is not very time-consuming and

an adequate number of Gauss points can be used to determine the stiffness (generally

three points suffice) Initial stress and other load matrices are similarly obtained

The particular isoparametric formulation presented in summary form here differs

somewhat from the alternatives of references 7, 8, 13 and 15 and has the advantage

that, because of its isoparametric form, rigid body displacement modes and indeed

the states of constant first derivatives are available Proof of this is similar to that

contained in Sec 9.5 of Volume 1 The fact that the forms given in the alternative

formulations have strain under rigid body nodal displacements may not be serious

in some applications, as discussed by Haisler and S t r i ~ k l i n ~ ~ However, in some

modes of non-axisymmetric loads (see Chapter 9) this incompleteness may be a

serious drawback and may indeed lead to very wrong results

Constant states of curvature cannot be obtained for afinite element of any kind

described here and indeed are not physically possible When the size of the element

decreases it will be found that such arbitrary constant curvature states are available

in the limit (see Sec 10.10 in Volume 1)

As in the straight frustrum element, addition of nodeless (enhanced) variables in the

analysis of axisymmetric shells is particularly valuable when large curved elements are

capable of reproducing with good accuracy the geometric shapes Thus an addition of

a set of internal, hierarchical, element variables

2 Nj A a j

j = l

(7.37)