Finite Element Method - Non - linear structural problems - large displacement and instability _11

Finite Element Method - Non - linear structural problems - large displacement and instability _11 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.

11

large displacement and instability

1 1.1 Introduction

In the previous chapter the question of finite deformations and non-linear material behaviour was discussed and methods were developed to allow the standard linear forms to be used in an iterative way to obtain solutions In the present chapter we con- sider the more specialized problem of large displacements but with strains restricted

to be small Generally, we shall assume that ‘small strain’ stress-strain relations are adequate but for accurate determination of the displacements geometric non-linearity needs to be considered Here, for instance, stresses arising from membrane action, usually neglected in plate flexure, may cause a considerable decrease of displacements

as compared with the linear solution discussed in Chapters 4 and 5, even though displacements remain quite small Conversely, it may be found that a load is reached

where indeed a state may be attained where load-carrying capacity decreases with

continuing deformation This classic problem is that of structural stability and obviously has many practical implications The applications of such an analysis are clearly of considerable importance in aerospace and automotive engineering applica- tions, design of telescopes, wind loading on cooling towers, box girder bridges with thin diaphrams and other relatively ‘slender’ structures

In this chapter we consider the above class of problems applied to beam, plate, and shell systems by examining the basic non-linear equilibrium equations Such considera- tions lead also to the formulation of classical initial stability problems These concepts are illustrated in detail by formulating the large deflection and initial stability problems for beams and flat plates A lagrangian approach is adopted throughout in which

displacements are referred to the original (reference) configuration

11.2.1 Geometrically exact formulation

In Sec 2.10 of Volume 1 we briefly described the behaviour for the bending of a beam for the small strain theory Here we present a form for cases in which large

Fig 11.1 Finite motion of three-dimensional beams

displacements with finite rotations occur We shall, however, assume that the strains which result are small A two-dimensional theory of beams (rods) was

developed by Reissner' and was extended to a three-dimensional dynamic form by Sirno.* In these developments the normal to the cross-section is followed, as contrasted to following the tangent to the beam axis, by an orthogonal frame Here we consider an initially straight beam for which the orthogonal triad of the beam cross-section is denoted by the vectors ai (Fig 11.1) The motion for the

beam can then be written as

4 , = - xi = xi 0 + AiIZI (11.1) where the orthogonal matrix is related to the ai vectors as

If we assume that the reference coordinate Xl ( X ) is the beam axis and X 2 , X 3 ( Y , Z )

are the axes of the cross-section the above motion may be written in matrix form as

A l l A12 A13 { ;;} = { ;} = { ;} + { t} + [&; ;I; h ] {E} (11.3)

where u ( X ) , u ( X ) , and w ( X ) are displacements of the beam reference axis and where

A(X) is the rotation of the beam cross-section which does not necessarily remain

normal to the beam axis and thus admits the possibility of transverse shearing deformations

The derivation of the deformation gradient for Eq (1 1.3) requires computation of the derivatives of the displacements and the rotation matrix The derivative of the

large displacement theory of beams 367

rotation matrix is given by213

Here we consider in detail the two-dimensional case where the motion is restricted to

the X-Z plane The orthogonal matrix may then be represented as (ey = p)

dimensional specialization of the theory presented by Simo and c o - w o r k e r ~ ~ ~ ~ ~ ~ and is

called geometrically exact since no small-angle approximations are involved The

deformation gradient for this displacement is given by the relation

Using Eq (1 0.15) ,and computing the Green-Lagrange strain tensor, two non-zero

components are obtained which, ignoring a quadratic term in Z , are expressed by

Exx = u,,y + 5 (u$ + w$-) + ZAP,x = EO + Z K b

(11.9) 2Exz = (1 + u , ~ ) sinp + w,, cosp = r

where EO and r are strains which are constant on the cross-section and Kb measures change in rotation (curvature) of the cross-sections and

A = (1+yx)cosp-w,xsin,6 (11.10)

A variational equation for the beam can be written now by introducing second Piola-Kirchhoff stresses as described in Chapter 10 to obtain

SII = f 0 (SEXX S,yx + 2SExz Sxz) d V - SIIext (1 1.11)

where SII,,, denotes the terms from end forces and loading along the length If we separate the volume integral into one along the length times an integral over the beam cross-sectional area A and define force resultants as

the variational equation may be written compactly as

(Sp T P + 6r Sp + SKb Mb) d X - 6IIeXt (11.13) where virtual strains for the beam are given by

SEO = (1 + u,x)Su,x + w,xSw,x

SKb = ASP,, + I'6p + cos P S U , ~ + sin ~ S W , ~

A finite element approximation for the displacements may be introduced in a manner identical to that used in Sec 7.4 for axisymmetric shells Accordingly, we

can write

(1 1.15)

where the shape functions for each variable are the same Using this approximation the virtual work is computed as

Large displacement theory of beams 369

where

(1 1.17)

Just as for the axisymmetric shell described in Sec 7.4 this interpolation will lead to

‘shear locking’ and it is necessary to compute the integrals for stresses by using a

‘reduced quadrature’ For a two-noded beam element this implies use of one quadra-

ture point for each element Alternatively, a mixed formulation where r and Sp are

assumed constant in each element can be introduced as was done in Sec 5.6 for the

bending analysis of plates using the T6S3B3 element

The non-linear equilibrium equation for a quasi-static problem that is solved at

each load level (or time) is given by

(1 1.18)

For a Newton-Raphson-type solution the tangent stiffness matrix is deduced by a

linearization of Eq (11.18) To give a specific relation for the derivation we

assume, for simplicity, the strains are small and the constitution may be expressed

by a linear elastic relation between the Green-Lagrange strains and the second

Piola-Kirchhoff stresses Accordingly, we take

Sxx = E E x x and Sxz = 2GExz (1 1.19)

where E is a Young’s modulus and G a shear modulus Integrating Eq (1 1.12) the

elastic behaviour of the beam resultants becomes

T P = E A F , Sp = K G A r and M b = E I K b

in which A is the cross-sectional area, I is the moment of inertia about the centroid,

and K is a shear correction factor to account for the fact that Sxz is not constant on

the cross-section Using these relations the linearization of Eq (1 1.18) gives the

tangent stiffness

where for the simple elastic relation Eq (1 1.20)

EA

and KG is the geometric stiffness resulting from linearization of the non-linear

expression for B After some algebra the reader can verify that the geometric stiffness

11.2.2 Large displacement formulation with small rotations

In many applications the full non-linear displacement field with finite rotations is not needed; however, the behaviour is such that limitations of the small displacement theory are not appropriate In such cases we can assume that rotations are small so that the trigonometric functions may be approximated as

equilibrium equations given by Eq (1 1.18) in which now

2 E x z = w , x + ,l3 = r

(1 + .,X) N q x w,x Na,x

large displacement theory of beams 371

This expression results in a much simpler geometric stiffness term in the tangent

matrix given by Eq (1 1.20) and may be written simply as

It is also possible to reduce the theory further by assuming shear deformations to be

p = - W , X (1 1.27) negligible so that from r = 0 we have

Taking the approximations now in the form

-

in which p, at nodes

The equilibrium equation is now given by

where the strain displacement matrix is expressed as

(11.28)

( 11.29)

( 1 1.30)

The tangent matrix is given by Eq (1 1.20) where the elastic tangent moduli involve

only the terms from TP and Mb as

Example: a clamped-hinged arch

To illustrate the performance and limitations of the above formulations we consider

the behaviour of a circular arch with one boundary clamped, the other boundary

hinged and loaded by a single point load, as shown in Fig 11.3(a) Here it is necessary

to introduce a transformation between the axes used to define each beam element and

the global axes used to define the arch This follows standard procedures as used

many times previously The cross-section of the beam is a unit square with other

properties as shown in the figure An analytical solution to this problem has been

obtained by da Deppo and Schmidt6 and an early finite element solution by Wood

Fig 11.3 Clamped-hinged arch: (a) problem definition; (b) load deflection

and Zienkiewic~.~ Here a solution is obtained using 40 two-noded elements of the types presented in this section The problem produces a complex load displacement history with ‘softening’ behaviour that is traced using the arc-length method described in Sec 2.2.6 [Fig 11.3(b)] It is observed from Fig 11.3(b) that the assump-

tion of small rotation produces an accurate trace of the behaviour only during the early parts of loading and also produces a limit state which is far from reality This emphasizes clearly the type of discrepancies that can occur by misusing a formulation

in which assumptions are involved

Deformed configurations during the deformation history are shown for the load parameter p = EZ/PR2 in Fig 11.4 In Fig 1 1.4(a) we show the deformed configura-

tion for five loading levels three before the limit load is reached and two after

Elastic stability - energy interpretation 373

Fig 11.4 Clamped-hinged arch: deformed shapes (a) Finite-angle solution; (b) finite-angle form compared

with small-angle form

passing the limit load It will be observed that continued loading would not lead to

correct solutions unless a contact state is used between the support and the arch

member This aspect was considered by Simo et aL8 and loading was applied much

further into the deformation process In Fig 11.4(b) we show a comparison of the

deformed shapes for p = 3.0 where the small-angle assumption is still valid

The energy expression given in Eq (10.37) and the equilibrium behaviour deduced

from the first variation given by Eq (10.42) may also be used to assess the stability

of equilibrium.' For an equ'ilibrium state we always have

conversely, instability by a negative value (as in the first case energy has to be

added to the structure whereas in the second it contains surplus energy) In other

words, if KT is positive dejinite, stability exists This criterion is well known' and of

considerable use when investigating stability during large deformation."," An

alternative test is to investigate the sign of the determinant of KT, a positive sign

denoting stability l 2

A limit on stability exists when the second variation is zero We note from

Eq (10.66) that the stability test then can be written as (assuming K L is zero)

S U ~ K ~ ~ U + s U ~ K ~ ~ U = o (11.35)

This may be written in the Rayleigh quotient form13

The limit of stability is sometimes called neutral equilibrium since the configuration

may be changed by a small amount without affecting the value of the second variation (i.e equilibrium balance) Several options exist for implementing the above test and the simplest is to let X = 1 + AA and write the problem in the form of a generalized linear eigenproblem given by

KT SU = AXKG 6~ ( 1 1.38)

Here we seek the solution where AA is zero to define a stability limit This form uses the usual tangent matrix directly and requires only a separate implementation for the geometric term and availability of a general eigensolution routine To maintain numerical conditioning in the eigenproblem near a buckling or limit state where K T

is singular a shift may be used as described for the vibration problem in Chapter 17

of Volume 1

Euler buckling - propped cantilever

As an example of the stability test we consider the buckling of a straight beam with one end fixed and the other on a roller support We can also use this example to show the usefulness of the small angle beam theory

An axial compressive load is applied to the roller end and the Euler buckling load computed This is a problem in which the displacement prior to buckling is purely axial The buckling load may be estimated relative to the small deformation theory

by using the solution from the first tangent matrix computed Alternatively, the buckling load can be computed by increasing the load until the tangent matrix becomes singular In the case of a structure where the distribution of the internal forces does not change with load level and material is linear elastic there is no difference in the results obtained Table 11.1 shows the results obtained for the propped cantilever using different numbers of elements Here it is observed that accurate results for higher modes require use of more elements; however, both the finite rotation and small rotation formulations given above yield identical answers

Table 11.1 Linear buckling load estimates

large displacement theory of thick plates 375

since no rotation is present prior to buckling The properties used in the analysis are

E = 12 x lo6, A = 1, I = 1/12, and length L = 100 The classical Euler buckling load

with the lowest buckling load given as a! = 20.18.14

11.4 large displacement theory of thick plates

11.4.1 Definitions

The small rotation form for beams described in Sec 11.2.2 may be used to consider

problems associated with deformation of plates subject to ‘in-plane’ and ‘lateral’

forces, when displacements are not infinitesimal but also not excessively large

(Fig 1 1.5) In this situation the ‘change-in-geometry’ effect is less important than

the relative magnitudes of the linear and non-linear strain-displacement terms, and

in fact for ‘stiffening’ problems the non-linear displacements are always less than

the corresponding linear ones (see Fig 1 1.6) It is well known that in such situations

the lateral displacements will be responsible for development of ‘membrane’-type strains and now the two problems of ‘in-plane’ and ‘lateral’ deformation can no

longer be dealt with separately but are coupled

Fig 11.5 (a) ’In-plane’ and bending resultants for a flat plate; (b) increase of middle surface length owing to lateral displacement

Fig 11.6 Central deflection w, of a clamped square plate under uniform load p;" u = v = 0 at edge

Generally, for plates the rotation angles remain small unless in-plane strains also become large To develop the equations for small rotations in which plate bending

is modelled using the formulations discussed in Chapter 5 we generalize the displace- ment field given in Eq (4.9) to include the effects of in-plane displacements Accordingly, we write

" = { ii) = { $,;;} -z{ eY(X/?}

where 8 are small rotations defined according to Fig 4.3 and X , Y, 2 denote positions

in the reference configuration of the plate Using these to compute the Green- Lagrange strains given by Eq (10.15) we can write the non-zero terms as

[2;;y}=[ u, y v,Y+;(w,Y)2 + v,x + W ; X w , y 1 - z [ ex,r :;: + OY,X } (11.41)

In these expressions we have used classical result^'^ that ignore all square terms involving 8 and derivatives of u and v, as well as terms which contain quadratic

powers of Z

Generally, the position of the in-plane reference coordinates X and Y change very

little during deformations and we can replace them with the current coordinates x and

large displacement theory of thick plates 377

y just as is done implicitly for the small strain case considered in Chapter 4 Thus, we

can represent the Green-Lagrange strains in terms of the middle surface strains and

changes in curvature as

= EP - ZKb (11.42)

u , y + u,x + W , x W g

where EP denotes the in-plane membrane strains and Kb the change in curvatures

owing to bending In addition we have the transverse shearing strains given by

= { -Ox -6Y + + W Y w+ 1 (11.43) The variations of the strains are given by

(1 1.44)

Using these expressions the variation of the plate equations may be expressed as

(6Kb)TSZdV-611,,t (11.46) Defining the integrals through the thickness in terms of the 'in-plane' membrane forces

s x x

T P = { :} =r2 -112 S d Z = r 2 -112 { S y y } d z ( 1 1.47) transverse shears

and bending forces

we obtain the virtual work expression for the plate, given as

[(SEp)TTp + S(I'S)TTS + c ~ ( K ~ ) ~ M ~ ] dA - SII,,, ( 1 1 S O )

This may now be used to construct a finite element solution

11.4.2 Finite element evaluation of strain-displacement

matrices

For further evaluation it is necessary to establish expressions for the finite element B

and K T matrices Introducing the finite element approximations, we have

-T

a, = [ u a 21, w, ( f i x ) , (d$),I = [u: w:]

-T

u, = [ii, G,] and w = [W, (ex), (ey),]

We here immediately recognize an in-plane term which is identical to the small strain (linear) plane stress (membrane) form and a term which is identical to the

Large displacement theory of thick plates 379

small strain bending and transverse shear form The added nonlinear in-plane term

results from the quadratic displacement terms in the membrane strains

Using the above strain-displacement matrices we can now write Eq (1 1 SO) as

6II = 66: SA (B:)TTp dA + 6w:

S A

(BL)TTs dA + 6W: (Bk)TMb dA - me,, = 0

(1 1.57) Grouping the force terms as

and the strain matrices as

the virtual work expression may be written compactly as

6II = 66: / A B:adA - bIIext = 0

The non-linear problem to be solved is thus expressed as

11.4.3 Evaluation of tangent matrix

A tangent matrix for the non-linear plate formulation may be computed by a linear-

ization of Eq (1 1.60) Formally, this may be written as

d ( 6 n ) = 66: SA [d(Bi)o + B:d(o)] dA - d(bn,,,) = 0 (1 1.62)

We shall assume for simplicity that loading is conservative so that d(611ext) = 0 and

hence the only terms to be linearized are the strain-displacement matrix and the

stress-strain relation If we assume linear elastic behaviour, the relation between

the plate forces and strains may be written as

DP 0

(11.63)