congruential formulation for geometrically nonlinear tl finite elements

NASA-CR-199064

CU-CAS-94..08

CENTER

FOR

AEROSPACE

A SURVEY

OF

STRUCTURES

THE

CONGRUENTIAL FOR TL

COREFORMULATION

GEOMETRICALLY FINITE

ELEMENTS

(NASA-CR-I99044) CORE-CONGRUENTIAL GEOMETRICALLY ELEMENTS

NONLINEAR

A SURVEY OF THE FORMULATICN FOR NONLINEAR TL FINITE

(Colorado

Univ.)

58

N95-322C3

Unclas

p

G3/61

by C. A. Felippa,

L. A. Crivelli

and B. Haugen

May

1994

COLLEGE

OF ENGINEERING

UNIVERSITY CAMPUS BOULDER,

OF COLORADO

BOX 429 COLORADO

80309

0060385

ml

r_

A Survey

of the

Core-Congruential

Geometrically

Nonlinear

CARLOS

A.

Department and

TL

FELIPPA,

AND

LUre

BJDRN

Boulder,

A.

Elements

CRIVELLI

Engineering

for Aerospace

University

Finite

for

HAUGEN

of Aerospace Center

Formulation

Sciences

Structures

of Colorado

Colorado

80309-0429,

USA

May1994

Report

Invited

survey

Computational and

article Methods

O. C. Zienkiewicz.

No.

contributed

CU-CAS-94-08

to the

in Engineering. To appear

inaugural Editors:

August

issue

of

Archives

M. Kleiber,

of

E. Ofiate

1994.

The research surveyed in this article was supported by the Air Force Office of Scientific Research under Grant F49620-87-C-0074, NASA Langley Research under Grant

Center

under

Grant NAG3-1273, ASC-9217394.

Grant and

NAG1-756, the

National

NASA

Lewis

Science

Research

Center

Foundation

under

A Survey

of the

Geometrically

Core-Congruential

Nonlinear

C. A. FELIPPA*,

TL

L. A. CRIVELLI

Formulation

Finite

** and

for

J.

Elements

B. HAUGEN

***

SUMMARY This article presents a survey of the Core-Congruential Formulation (CCF) for geometrically nonlinear mechanical finite elements based on the Total Lagrangian (TL) kinematic description. Although the key ideas behind the CCF can be traced back to Rajasek_aran and Murray in 1973, it has not subsequently received serious attention. The CCF is distinguished by a two-phase development of the finite element stiffness equations. The initial phase develop equations for individual particles. These equations are expressed in terms of displacement gradients as degrees of freedom. The second phase involves congruential-type transformations that eventually binds the element particles of an individual element in terms of its node-displacement degrees of freedom. Two versions of the CCF, labeled Direct and Generalized, are distinguished. The Direct CCF (DCCF) is first described in general form and then applied to the derivation of geometrically nonlinear bar, and plane stress elements using the Green-Lagrange strain measure. The more complex Generalized CCF (GCCF) is described and applied to the derivation elements. Several advantages of the CCF, notably the physically

of 2D and 3D Timoshenko beam clean separation of material and

geometric stiffnesses, and its independence with respect to the ultimate choice of shape functions and element degrees of freedom, are noted. Application examples involving very large motions solved with the 3D beam element display the range of applicability of this formulation, which transcends the kinematic limitations commonly attributed to the TL description.

1.

There is an elegant cal finite elements as the

Core-Congruential

by Rajasekaran pioneer

work

nonlinear metric

derivation

Murray

Marcal,

analysis.

for the

equations.

This

or CCF,

1 in 1973,

and

element

expressions

Formulation, 1987

and

(TL) formulation of geometrically nonlinear mechanilittle attention in the literature. This will be referred to

Formulation,

of Mallet

finite

governing

In

Total Lagrangian that has received

INTRODUCTION

from

2 as well

3 The

stiffness work

evolved

in the

originated

the

and

previous

of Reference

that what

The

analysis

as Murray's

discussion

matrices

sequel.

appear

here

presented

reinterpretation work

4 provided

levels

the Direct

of the

in geometrically

1 by Felippa

at various

is called

key concepts,

of the

Core

paradiscrete

Congruential

or DCCF.

a course of several

in nonlinear elements

finite

DCCF.

by

first

author

the

Uni-

Norwegian

Institute

of homework

presented

* Department of Aerospace Engineering Sciences and Center for Aerospace Structures, versity of Colorado, Boulder, CO 80309-0429, USA "" Hibbit, Karlsson & Sorensen Associates, 1080 Main St., Pawtucket, RI 02860, USA The

Preparation

the

and

Engineering,

the

offered

assignments

*** Division of Structural helm, Norway

using

elements

of Technology,

N-7034

Trond-

feedback

from

Subsequently

students

in this

Crivelli's

doctoral

three-dimensional large

nonlinear

rotations.

frontiers This

explained

in more

A lesson should

was

gained

from

be applied

terms

arising

methodology

gave

Both

DCCF

and

core

equations

aspects,

Sections

amenable

to what

well

the

10, and

same

as subsequent

3 through direct

called

To

7 focus

on are

with

and

the

transformation

However,

equations

focusing

the

to physical

on

the

of application

essential

to elements

is discussed

3D beam

CCF

or GCCF.

those

GCCF

6 and

of additional

philosophy.

while

to 2D and

Crivelli

rotations, That

CCF,

conquer"

The

of freedom.

and

examination

of freedom.

Examples

beyond

Felippa/

3D finite

transform

presented.

applications

with

exposition

DCCF.

degrees

by Felippa

and

of a

arbitrarily

formulation

rotational

Generalized

that the

the

this

systematic

degrees

the

steps

treatment

illustrated

the

"divide

simplify

pushed article

material.

development

of undergoing

with

dealing

allows

systematic

the

capable

by Crivelli

to physical is here

share

element

when

to streamline

in the

application

paper

that

transformations

CCF

in a survey

is that,

fashion

in complexity.

to the

through

reported

helped

element

by a TL

research

GCCF as

vary

this

rise

this

in a subsequent

in a staged

in the

freedoms

summarily

the

beam

by

impassable

detail

offerings

thesis 5 used

posed

deemed

development

follow-up

Timoshenko

Challenges

hitherto

and

in Sections

8

elements.

REMARK 1.1. Several authors have expressed the belief that the approximation performance of TL-based elements degrades beyond moderate rotations, and an updated Lagrangian or corotational description is necessary for handling truly large motions. For example, in 1986 Mathiasson, Bengtsson and Samuelsson s concluded that "The TL formulation can only be used in problems with small or moderate displacements." More recently Bergan and Mathisen 9 voice a similar opinion: "it is commonly known that in a step by step TL formulation artificial strains easily arise in beam elements due to nonhomogeneities in the displacement expansions in transverse and longitudinal directions." Our experience shows that such limitations are not inherent in the TL description but instead emerge when a priori kinematic approximations are made to simplify element derivations. The 3D beam element just cited exhibits computational and approximation performance Lagrangian

for very large rotations comparable descriptions while retaining certain 2.

2.1

Basic

The

original

matrices tern

to those based on the co-rotational advantages listed in the Conclusions.

and

Updated

of TL

stiffness

OVERVIEW

Concepts development

of the

for geometrically

CCF

nonlinear

was

concerned

analysis

with

through

the

the

construction

congruential-transformation

pat-

$*

Klevel where

S is the

freedom of v,

core

stiffness

matrix,

v, G a core-to-physical-freedom V0 the

governing

appropriate

equation

level

= Iv,

GTslevetG

K the physical

stiffness

transformation

reference

integration

at which

the stiffness

dV,

volume, matrix

(2.1) in terms

matrix and

assumed

in which

is used.

of the

nodal

degrees

of

to be independent "level"

identifies

the

The

three

variational

(level 1), and "secant-stiffness" The

core

levels

of interest

in practice

first-order incremental equilibrium are also used for level 1, and

stiffness

matrix

is expressed

in terms

terial point. Displacement gradients g make strains because for elements with translational expressed

linearly

in terms

(2.1) for all levels. CCF. The

qualifier

grees

such

of freedom.

2.2

Direct

The

basic

Generalized

of a finite

v as g = Gv,

shape

the

the

of

by

the

that

purview

S lewi

functions,

element

than finite they can be

a property

fall under

of independence

CCF,

mathematically

Congruential Transformation

Equations

Equations

view

displacement element

needs

of the

with

and

validates

respect

choice

congruential

Direct

to dis-

of nodal

de-

transformation

volume.

expressed

through

(2.1),

may

be dia-

model)

is linear,

Stiffness

(1) Core

Secant

Stiffness

(1) Core

Internal

this

Tangent

g)

equation

in use.

expressions but

this

assumption

relation

the

precise.

physical

If the

DOFs

transformations

within Internal

at level

is not

do not

the

displacement

sketched

above

in the Direct

not

"core

force

and

energy diagrams

Core

gradients are

3

box"

node

between

core

displacements

depend

v

on level:

Physical

Energy

Stiffness

Physical

Secant

Stiffness

Physical

Internal

denote

secant level

to reduce

more

node

the are

0) may

two

level

alternative

also be expressed

clutter.

displacements but

Stiffness

variational

Formulation,

complex

Force

Tangent

stiffness (level

Congruential

g and only

relation

==_

the

1. The

shown

we obtain

between

(the

Physical

governing

governing-equation

transformations

more

Stiffness

annotated

mentioned

Equations

Transformation I Congruential Equations

Force

numbers

ways,

these

==_

diagram,

in several

and

Physical-DOF Stiffness

==_

to be rendered

gradients

Energy

(2) Core

the

a better choice of core variables degrees of freedom (DOFs)

Core Stiffness

(0) Core

If the

at each

CCF

of the

panoramic

(the

of the

gradients

as

this

DOFs

and

equilibrium

ma-

displacement

elements

over

0), force

of the

is introduced

integration

(level

and

geometry,

a dependence the

schematics

grammed

But

and

goal

as element

Such

in (2.1)

the

such

energy

(level 2). Qualifiers "residual-force" "tangent-stiffness" used for level 2.

displacements

below,

emphasizes

decisions

indicated

In

As discussed

"core"

cretization

of node

are:

depend

Under

the

afore-

or DCCF. v is nonlinear, on variational

level and possibly the expressionform used within a level. This complication

arises

elements

considered.

with

It gives Two

rise

rotational to the

variants

nonlinear are

still

(0) Core

may

be

in the

plates

If the

equations

following

shells

are

when

or GCCF. relation

do vary

between

with

level

g and

but

v is

in principle

diagram.

Physical

Energy

Stiffness

Physical

Secant

Stiffness

=:_

Equations T-Congruential Transformation

Stiffness

and

Formulation,

S-Congruential Transformation

Force

Tangent

as beams,

distinguished.

transformation

Stiffness

Internal

such

Congruential

Stiffness

Secant

(1) Core

the

as illustrated

Energy

(1) Core

Core

GCCF

algebraic,

possible

of freedom

Generalized

of the

but

(2) Core

degrees

Physical

Internal

Physical

Equations

Force

Tangent

Stiffness

i

Here

"T-Congruential"

and

"S-Congruential"

tial" and "Secant-Congruential," Section 8. If the

relation

differential

between

form,

generally

exist,

the and

g and

(0) Core

Energy

Stiffness

(1) Core

Secant

Stiffness

(1) Core (2) Core

These

two

denoted

by

Internal

Stiffness

variants

of the

acronyms

tinction between quantities, such The with of the

original

must

GCCF

are and

AGCCF and as the physical

CCF,

development

or DCCF, and

of the

only

of the

upon

in

in non-integrable

preceding

Physical

diagram

Algebraic

DGCCF,

GCCF

GCCF

and

respectively,

in the

it makes with the

no sense latter.

outlined

Sections

Internal

Physical

in the

configurations.

of the

Congruen-

do not

be truncated:

called

CCF

is sufficient.

application

be expressed

Equations"

DGCCF is that secant stiffness,

translational-degree-of-freedom

"Tangent is elaborated

Equations

AGCCF

development

can

for

a distinction

T-Congruential Transformation

Force

Tangent

and

Transformation

diagram

abbreviations

Such

v is nonlinear

"Secant the

are

respectively.

3 through

to Section

Tangent

such focus

sequel.

8 and

following

and

main

dis-

about

missing

on

elements

focused

that

GCCF The

to talk

elements on

Stiffness

Differential

Introduction For

Force

the form, ones.

Direct leaving

form the

2.3

The

CCF

The

CCF

derivation

work. In the

initial

key

spect

matrices

goal

is to try

is in fact

is linear,

which

nonlinear

because the

that

the

and are

node

stages

that

Decisions to final

such

are

the final

the

differences

discrete

equations

common,

a priori how

transformed

may In the

rotational

coined.

Complete

gradients

g and

if the

level.

Such

which

be done

are

v

relation

is

dependencies

characteristic

of

GCCF

areas

parametrizations

the

elements beam

and

transformation

through

nodal

for finite

as 3D

into

or volumes

to

i.e.

transformations

such

is decomposed

domain

DOFs,

for simple

multistage

Differential lines,

to physical directly

particular,

phase into

staged

material

and

effect the

The behavior,

shells.

kinematic

degrees

rotations

of freedom.

may

be deferred

displacement

the

origin

on

retention

subproblems

of each or that

conventional

order

is maintained

in the

"thrown

axial

of freedom

it is quite

becomes

applied of various

degrees

formulation

approximations

terms

DCCF

approximations

rotational

situations

Lagrangian throughout,

1 for the

analysis with

Total

difficult

into

virtually

the

to pot,"

impossible.

deformation

of a spinning

buckling?

recommended

decisions

elements

innocuous

conventional

in Appendix

nonlinear

of complex

of higher

more

exactness

This is shown

of seemingly testing

the

If kinematic

In the

for

core equations

sequence

and

in structural

neglect

torsional

approach

CCF

in geometrically

exhaustive does

the

elements?

especially

the

affects

assumptions.

testing

are

element

are identical. But

plates

a posteriori

3D beam

finite

elements.

as beams,

Sample:

between

of nonlinear

are

assess

The

stiffness terms,

ones.

particles

of specific

was

re-

functions,

DOFs)

tempered

s_iffness

forms

require

the

to be

with

shape

displacement

has

tangent

transformation

link

as possible

core

For

in (2.1).

geometry,

term

between

transformation

"bind" choice

core

that

S t''_t

of rotational

the

as internal

particles.

GCCF.

complex

the

eventually

as the

to continuum

and

case

progressively

formulation

such

for

case

goal

at the

term

phase.

as well

to individual

frame-

stages.

What

kinds

these

for elements

and

The

of the

The

fashion In this

constraints,

relation

geometric

purview

displacements.

recommended elements.

if the DCCF. arise

matrices

pertain

as element

the

outlined

stiffness

as independent

such (in

the

by a transformation

by the

independence,

may

phase,

in multistage

shell

the

complementary

fall under

and

this

achievable

transformation

element

core

decisions

reflects

followed

vectors

equations

dependencies

so-called

elements

tangent and

such

of freedom

characterizes

phase

represented

To emphasize

independence

naturally

collectively

to make

degrees

parametrizations.

create

are

and

matrices

discretization

of nodal

a core

secant

These

Conquer

equations

phases:

energy,

they

to finite-element

selection

In

core

and

element

two

are obtained.

stiffness

The

finite

through

phase

vectors

Divide

of the

It proceeds

force the

Philosophy:

the

gradients term

can

dropping isolate

GCCF

are physically

the

and be

a better clearly

prestresses.

In the

accurately

traced,

made.

This

physics

modeled 5

permits

transparent,

process

and can

by specific

be

control

displaying ensuing on terms.

such

effect

of

transformation

that

aided

over the

basis by

informed

computer

by

From

this

discussion

ation

and

testing,

clean

separation

timated,

it follows the

most

physical

Mallet

rically

nonlinear

kinematic tem v.

under

dead C.

The

are The

system the

has

r = OJ/Ov,

current used Mallet

finite

the

Marcal

conjugate

symbol

rather

and

loads

up

virtual the

the

total

OJ

N1 that

K0 and

is the N2

linear

are

depend

linearly

matrices

were

survived; functions.)

presently

Five

years

matrices from

said

later that

the

"core"

specific

"to

function

Rajasekaran

and

in the

stiffness

with

the

the

latter

are identified

by the

gradients, elements

uniquely

determined.

freedom.

They

This

was

partly

those

equations,

by Felippa, here j=

used

4 who

1

_v

T-_U

_

equi-

(3.1)

that

down

in the

v+(p°-p) 6

old

the

main

are

unique

expression

and

the they N2

idea only valid

T

v,

N not

of element

shape

structure

of the

chose

CCF.

N1 and

to start in terms Working

N2 are

not

degree

of

for a single for arbitrary

of Reference

compact

The has

expressed

of the

matrices

v.

notation

matrices

critically

N1

whereas

configuration,

displacements

(This

stiffness

and

reference

investigation

discussion

general

(3.3) configuration,

node

more

nonlinear

(3.2)

= 0.

to identify

to K,

a general

in a more

(5 prefix.)

pTv

at the

expressions.

as written

however,

usual

are

(force-balance)

reference

on the

In

so laid

- Ap

the

1 examined

the

variations

p = 0,

evaluated

corresponding that

v -

v-

Av

at

also

foregoing

(3.1)-(3.3)

present,

rewritten

evaluated

in doing

of the

as follows:

+ _N2] 1

+ _N_]

equations.

showed

Indeed

did not done

and

they

residual

forces

1

+ _N,

Murray

the

equations

difference node

variation

ones;

energy,

con-

in array

is the

associated

commonly

matrices

that

increment

respectively,

above

W

residual

matrices,

N is most

-- U -

The

equilibrium

in array

collected

W = pTv.

potential

in the

J

of v, are

2 actual

matrix

repeat"

collected

sys-

Co to a current

of Reference

quadratically,

symbol

appear

of displacement with

and

of freedom

(TL)

structural

potential

1

stiffness

is the

Lagrangian

spirit

[K0

stiffness

nonlinear

CCF

be underes-

for geomet-

of a static

configuration

independent

Ar = [Ko + N1 -}- N2] Here

Total

the

__

0v

the

model

degrees

= _vl T [Ko + _N1 ,1

r -

evalu-

analysis.

with

incremental

J=U-W

the not

nomenclature

on

element

forces,

A denotes

than

based

reference

energy

the

keeping

expressed

equations,

a fixed

potential

for

should

in nonlinear

a standard

displacement

from

U and

(In

Section

and

librium

a discrete,

a total

configuration. in this

Consider

nodal

factor

to success

to establish analysis

with

be claimed

of this

key

structural

measured

can

development,

BACKGROUND

2 attempted

energy

and

is the

of element

that

importance

HISTORICAL

virtual-work

strain

standpoint

element

loading

Displacements

between are

finite

The

transparency

Marcal

description.

figuration p.

and

the

advantage

effects.

3.

In 1968

from

significant

of physical

because

that,

1 considered

elements. again

form: (3.4)

r

= K_v+

pO _ p = f_

Ar = KAv-in which K"

and

and

the

K denote

secant

other the

notation

in the

prestress

force

was omitted

in that while

important

secant denoted

In linear

problems

finite

which

secant

problems

(R1)

K _ is symmetric.

(a2)

The

reference

(R3)

The

finite

(a4)

The

transformation

terminology,

strain

TL

tinuum

be noted

finite

elements

material strains and the

current

strains

Xi The

state

and

xi

ei (i = 1, 2,...

stiffness

but,

The

and

tangent stability

Krv

+ p0 ) is

application

matrices

as shown

Such

core

physical

and

restrictions

per

coalesce.

in the

parametrized

that

displacement

(R1)

restriction

se

next

But section,

expressions

were

gradients.

freedoms and

(R4)

(R2)

is the

is linear. altogether,

condition

and that,

the

with

other present

DCCF.

STIFFNESS

EQUATIONS

Motion structure u.

under

These

We confine

the

linear

rotations. are

limited

p0 is free

and

form

for

free.

at this stage.

displacement

of strain

differ

coefficients.

current

within

three

in the

displacements

large

used

two.

is quadratic

nonlinear

configuration

coordinates

the

stays

arbitrarily

matrices

Co to a variable

is implied

behavior but

dimensions. Let

finite

configuration

the

vector.

methods

internal-force

other

(Energy are

is stress

force

enjoys

K U,

In addition,

solution

for the and

symbols

configuration internal

or in the

is stress

CORE

geometrically

° is the

Here

respectively.

such

Introduction.)

reference

stiffness

is used.

restrictions:

of Particle

undergoing

reference

the scalar

following

the

Description

A conservative,

do

between

characterizes

in the

energy

= K0

eliminates

It should

itself

The

measure

4.

4.1

(by

4 --

matrices,

K s because

noted

f = K"v+p

configuration

treatment

two selectively.

only

the

stiffness

if the

as source

arbitrary

4 under

tangent

of Reference

in incremental-iterative

stiffness

not

that

by K e and

methods.

involve

by Felippa

following

4 and

=0, than

course

K v = K r = K

may

K"

and

vanishes

importance

K v and

rather

element

fundamental

in nonlinear

The

not

in pseudo-force theoretical

given

energy,

discussion,

the

has

--

are

vector,

is of course

analysis, but

the

stiffnesses

purposes

stiffness

of this paper

Ap

p = 0,

field

rid),

our

attention thus

in the

Cartesian

current

ns)

in an array 7

No to the

implying

points

are

at a particle collected

are measured C.

respectively,

components

is viewed

configuration range,

to a fixed

loading

displacements

Corresponding

referred

(i = 1,...

elastic

dead

coordinate

where

from

a fixed

discretization

into

case

small

or particles

in which

the

deformational in the

system

nd is the

as a con-

number

reference and

have

of space

ui = xi - Xi. configuration

e, and

let

the

be

characterized

corresponding

by conjugate

ns

stressesbe the

elastic

si (i = 1, 2,... us), stress-strain relations

collected in an array are written

si = s 0i + Eijej, 0

where

s i are

called usual

stresses

prestresses) manner.

Let

if,///,

W,

the

particle

a dead-loading

body

are

first

force

density

total

strain

energy

=

introduce

identified array

the

Following

to the

oi

hi and

metric. the

ng displacement ng)

Hi

are

original

for the

that

The

eTs0

+ leTEe.

convention

+Ee,

S0

(4.1)

remain

if ei = 0, also

square

p, f and

of energy

array

in the

r, respectively,

densities,

strain

at

whereas

energy

density

fit is can

be

Energy

As noted degrees

counterparts"

below

1 and

through 1

T

strain

matrix

measure.

in Section

arranged

relations

structure

These

This

description

are

that

of the

the

--

subsequently strains

ei are

form

(4.3)

n,

ng x ng, respectively, that

volume:

in a one-dimensional

4 assume

i = 1, 2,...

assumed

the

in a TL

= Oum/OX_,.

ng x 1 and

1'4 it was

over

be expected

Felippa

Hi g,

of dimension

except

as can

gmn

(4.2)

(4.2)

can be conveniently

+ _g

References

--

Murray

gradients

arrays

integrating

gradients

and

Green-Lagrange

3, will be enforced

as

summation

as a ns x ns

of u.)

=

by

so they

Rajasekaran

displacement

In the

case

4.2

s=

(stresses

meaning

½eiEijej

+

taking place geometry.

ei = hTg where

the

U is obtained

as gi (i = 1, 2,...

g.

linked

the

of J, U, W,

independent

eis

U = fv0 U dV; the integration over the reference configuration Next,

or

,

arranged

analogues

acquire

Using

as l.t

The

moduli

the

three

= Eji

configuration

elastic

T denote

(The

Eij

reference

Eij

fit, _ and

level.

expressed

in the and

with

s.

Hi

is independent

restriction,

labeled

with

Hi

sym-

of g, which (R3)

is

in Section

4.5.

Variations

previously,

for deriving

of freedom.

On

of (3.4)-(3.6),

core

equations

substituting

(4.1)

in which

fl=LI_W=7

v has

we regard and

become

1 gT,-,U _

(4.3)

the into

displacement (4.2)

we obtain

gradients the

g "core

g:

g + (_0

_ fit)Tg,

(4.4)

0K T

__

0g

_Srg+fit°-_=_-fit

=0,

(4.5) (4.6)

_T=S_g-Afit=0. Here which

S U, S r and is independent

S denote

the

energy,

of g, is the

core

secant

and

counterpart 8

tangent of p0.

core

stiffness

matrices,

and

fit0

With this notation the first and second variations of the strain energy density can be expressedas _U = _gT(SUg t?H These

= 6gWSr

variational

linearity below,

6g + 6gT_Srg

equations

restriction

(R4)

+ :1gT_sU

+ _0)

implicitly holds,

the

g = 6gT (Srg

+ (_2g)T@ determine

term

in

_. _gT_,

(4.7)

(4.8)

= _gT S 6g + (_2g)W_.

S r, •

_2g

+ _0)

and

drops

out

S from

S U and

as explMned

If the

@0.

in the

Remark

and 5g.

--- 6gTS

62_

(4.9)

REMARK 4.1. If g = Gv with G independent of v, 8Zg = G62v = 0 because v are independent variables. On the other hand, if displacement gradients are nonlinear functions of node displacements

expressable _gi

as gi = gi(vi),

c

(_gi = _vjOVj

then

_2gi =

= G 0 8vi,

°q2gi

_fvjSv_ + ¢3gi _22_0

Ovj,gv-'-"_

Thus _g is still (_ _v but/52g = (F _v)/_v, where F is a cubic array. is taken into account in the GCCF discussed in Sections 8-10. 4.3

Parametrized

For convenience i,j

= 1,...ng

(4.10)

-_v o/'vj = F_jk,_v_,%k. The

presence

of the term

/_2g

Forms introduce

the

following

ng x n a matrices

(with

summation

convention

on

implied): S0 =Eij

hihj,

Sx = E 0 higTHj,

S_ =Eij

(hTg)

H i, (4.11)

S2 = E 0 Hi ggTHj, in which gTHig. stiffnesses

parentheses It may and

are

be then prestress

used

to emphasize

verified vector

S_ = Eij (gTHig)

that,

the

if assumptions

in (4.6)-(4.8)

H j,

grouping

of scalar

(R3)-(R4)

possess

the

general

quantities

of Section

3 hold,

such the

as core

form:

1 1 0 + sT) + (1 -_)s_ + i_s_ + i(1 - _)s_ + _,H_ 1 0 =S0+ + _,)H,, _(s, + s_) + (-_- _)s_ + ¼_(s_- s_) + :(_, 1 =So+ ½_(s, + s_r) - _s; + i_s_ - ¼(1+ _)s_ + _,H,, 1 * 0 s_(¢,_) = So+ ½Sx + CS_ r + (1 - ¢)s_ + ¼(2- ¢)s_ + _¢s_ + _H, 1 0 =So+ r +(½ -¢)s; + ¼(_- ¢)s_ + ¼(¢- 1)s_ + _(_, + _,)H,, ½S1 + Cs_ =So+ '(2 - $)(S2 s, + ,s, _- CS; + : - S*) +siHi,

sU(a, 3) = So +

S=So+

1oz(Sl

S1 +S T+S1

*

1

*

0

+S2+_S2+siHi=S0+S_

+S T+S2+s_H_,

o = sOhi. (4.12)

i

Here

a,

/3, ¢ and

independent

¢ are

of them.

arbitrary

scalar

coefficients

bi is defined

in (4.18)

originally

given

by Felippa

enforced.

Note

that

The

"repeatable ¢ = 1/2,

core

counterparts

interest.

These

respectively. If the

secant

The

Generalized

stiffness of Mallet

the

combinations

N2,

respectively. are

Sl

the

+S

is required

is called combination

the

name

and

Sl + S T + S_ But

principal

core

this

following

S2,

Srg

are

=

tangent

no longer

if ¢ = 1/2. if c_ =/3 1

and

= ¢ = 2/3

*

$2 + _S 2 become

observation

S0

+

has

largely

the

historical

SD,

(4.14)

material

is S = So + So

core

SM

3 are

those

= siHi.

initial-displacement,

=

than

combinations:

SM

principal

general

in Section

are obtained

Marcal

for downstream

the

noted

more

and

= S_ + _S 2* + s°Hi

stiffness

are

symmetric

Sc

of the

T+

(4.12)

S r becomes

=

S

receives

gTSUg

(4.13)

(R1)-(R2)

SD

tangent

CCF

expressions

and

versions

Section 8, Sc = sini SGp. In this case the

core

relevant

core

The

case

physically

core

that

= sibi,

restrictions

(3.1)-(3.3)

of N1

the

below.

4 because

in which

More

are

the

forms"

and

sense

In fact, = S_g + _o

where

in the

+

+ Sc

element

and

geometric

stiffness,

= SM + So. development

geometric

stiffness

and

as explained

in

is denoted

by

(4.15)

SGp,

stiffness.

REMARK 4.2. Finite element practicioners may be surprised at the nonuniqueness of S v and S r. It appears to contradict the fact that, given two square matrices Aa and As and an arbitrary nonzero Aa and

test vector x, Alx = A2x for all x implies A1 = As. But this is not necessarily true As are functions of x. More precisely, the energy core stiffness is not unique because gT(s,--Sl)g=0,

and

the secant

core stiffness

gT(sT--S;)g=0, is not unique

(4.16)

because

(sT - S;)g = o, Adding 6/4 and tangent is not a

g T ( S _ _ S;)g=0,

if

- S;)g = o.

(4.17)

"gage terms" such as those of (4.17) multiplied by arbitrary coefficients does not change consequently the secant stiffness acquires two free parameters. Uniqueness holds for the stiffness because the test vectors are the virtual displacement gradient variations, and S function of 6g.

REMARK 4.3. Because of (4.16), an additional free parameter appears is allowed. If symmetry is enforced the first two gage expressions must gT(s, + ST -- 2Sr)g = 0.

10

in S U if unsymmetry be combined to read

4.4

Spectral

There

is a more

as well and

Forms compact

alternative

as implementational

expression

advantages

of the

at the

cost

core stiffnesses of some

that

offers

generality.

theoretical

Define

vectors

bi

ci as ei=cTg,

Then

the

spectral

(4.21)

with

the

ci=hi+_Hig,

forms

(so

spectral

called

Oei = hi + Hig. 0--g

bi=

because

decomposition su(1,

1

of the

formal

of a matrix

1) = S U

similarity

as the

sum

(4.18)

of equations

of rank-one

(4.19)-

matrices)

are (4.19)

= Ei.i cic T + si0 Hi, a=13=l

r

St(0, S

r

(_,1)

1

that

S r

:

S

Note

=

S r (_',1 1) is symmetric

stiffnesses,

compactness

REMARK

4.4.

I¢=1,_b= 1 = Eij

Eij

bib T

but

is paid

The foregoing

0

(4.20)

0) = S I_,=,_=o = EO bi cT + si Hi,

S"(0,

in terms

relations

+

siHi

CiC

T

+ :(si

=

SM

+

0) is not.

of settling

1

(4.22)

SG.

It is seen

that

for specific

may be easily

verified

(4.21)

0)Hi,

+ si

for

energy

and

secant

coefficients.

by noting

that

1 1 E,_c,cT= So+ _(sl + sT) + _S_, 1

1

Eij blc T = So + _S1 + S T + _S2, Eij bib T = So + $1 + S T + $2,

E,, 0g

and seeking 4.5

c, \ 0g /

these

patterns

Generalization

If the

Hi

Lagrange's because variational

depend are

of the

used,

on

equations

+ \ 0g/

in the general H(g)

g,

as it generally secant

and

of first (4.7)-(4.8)

and

* 1 * S 1 -]- 782

parametrized

to

the

presence

=

tangent second

happens stiffness

if strain

measures

core equations expressed

(si

-

si o )Hi,

(4.13).

of Hi.

other become

The

changes

than more

Greencomplex

in the

core

as

((s"+gr)g+ .0 + S0) = 6_u = _gr(S + g)_g + (6_g)r(v + _). 11

:

EijejHi

expressions

g-derivatives

can be succintly

(4.z3) :

+

(4.24)

(4.25)

where _r, _ and _ are additional core terms that arise on account of the dependenceof the

Hi

The

on g.

parametrization

sures

and

of interest

problems. nonlinear

efficient

in practice,

Such topics finite element

notably

the

marily

core

determined

energy

density.

actual

structure

Several

cases

developed develop these 5.1

This

are

2-node

are

in 3D

out

For

systems

motion

force

the

bar

or 3-node

of the

strain

mea-

presently

projects

particle,

open

in advanced

g that

their

are retained

mathematical

the

basic

commitment core

form

is pri-

in the

strain

idealization

moving

in 3D space.

steps.

to

equations

curved

7, 9 and

of the

may

ones.

The

specific

Some

core

expressions

elements,

only

be subsequently specific

elements

to

a

used

to

based

on

is due

to

10.

and

displacement

components

rather than {Xl,X2,X3}

to this

Co the bar axis.

local

The

nd = 3, ns = 1 and

bar

system,

g2

motion

measure

0,{..+7

we adopt

\0X]

+k,

the

energy

contribution

3. To simplify

be

denoted

node

by

subscripting,

{X, Y, Z},

Cartesian

of a particle

initially

system

at

X

{X, Y, Z},

is defined

uY = uy(X) and uz - fiz(-_'). definition of nonlinear strains are

=

0fiy/0X

{x, y, z}

and {ul,u2,u3}, respectively.In

to a local

g3 strain

ng will

is referred

the

only

, {Xl,X2,Z3}

_x = fix(-_), intervene in the

g =

=

are

for term

of s, e and

in Sections

We have

the displacement components displacement gradients that

e=el=

strains,

of an individual

to illustrate

not

example

to a bar

the reference configuration with ._ located along the

As uniaxial

several

EXAMPLES

a byproduct

below

elements

stress.

reference

candidates

of components

do

derived

belongs

and {ux,uy,uz}

With

midpoint

for

Space

the longitudinal Cartesian

the

terms

component.

examples

straight

be good

is in turn

worked

model.

particle

choice

choice

and

STIFFNESS

reflect

or structural

equations Bar

The

by the

in these

mathematical

CORE

equations

of such

logarithmic

would in fact courses.

5.

Because

characterization

The

+ ½

92 ga

three

(5.1)

.

z l Ok Green-Lagrange

OX]

(GL)

axial

j=g,+5

+\OX]

strain,

defined

0 0

1 0

12

92 ga

= hTg

as

l+g_+g_)

[il { {gl} [1°i1{ } 1 g2 g3

by

+ _

g.

(5.2)

Thus for this choice of strain, h Tmatrix.

The

stress. the

The

conjugate

stress

stress-strain

reference

Because

H

directly

use

and

sl

of g,

spectral

-

0

to form

the

core

{

_g2

1

"_- } gl

b -

,

}

inserted

into

the

spectral

forms 1

1) = Ecc T + s°H

axial

coordinates

the

g2 l+gl} ga

axial

stresses

in

modulus.

in local

bl =

(PK2)

PK2

we can

vectors

,

(5.3)

yield 2

(1 + ggl) sU(1,

s are

construct

½ga which

3×3identity

E is Young's

stiffnesses First

Histhe

Piola-Kirchhoff

s o and and

(4.19)-(4.22).

1 =

-

second

where

respectively,

expressions

C _-_ C I

0] andH1

s is the

is s = s o + Ee,

configurations,

is independent the

measure

relation

current

h T = [1

1

1

½g3(1+

792(1 + 791) 1

= E

2

1

1

7ga

1

1 0

.._ $o

yg2g3 2

7ga

symm

, (5.4)

1

(1 + gg_)

2

1

1

St(½, 1) = Ecc r + s'_H = E

1

½ga(1 +

gg=(1 + 2

)

_gl

1

_

¥g2g3 1

1 0

s rn

2

7ga

symrn

(5.5) (1 + gl) 2 S = Ebb

T + sH

g2(1 + gl)

= E

ga(1 +gl)

g_ symm

In equation clean

(5.5),

separation

5.2

Plate

example

The

by {X,Y}, element

two

components

arranged

to move

geometric

and ux

strain

Consequently

a particle in its

reference {ux,uy}

displacement

to the

GL strain.

and

we consider

Cartesian

{x,y}

The

g_

o + s) = s o + ½Ee

_(sl

0 1 0

is the

average

(initial-stress)

1 0 0

or "half-way"

stiffnesses

should

0 0] 1

.

(5.6)

stress.

The

be noted.

Stress

constrained

midplane.

tribute

-_-

material

in Plane

As second brane),

sm into

+s

g_g3

= ux(X, energy

plane.

As

system

and

rather

field

that

than

and

and four

uy

usual

displacement

particle = uy(X,

displacement

nd = 2, ns = 3 and

to a plate

we consider

{XI,X2},

of a generic Y)

pertains

{xl,x2} Three

gradients

ng = 4. The

only

the

components

originally Y).

in plane

and

appear four

motion

(memof the

will be denoted

{ul,u2},

at (X, Y) in-plane

stress

respectively.

is defined PK2

in the

displacement

by the

stresses

con-

corresponding gradients

are

as

g2

Our/OX

g = {glga } = {Oux/OX} Oux/OY g4 Ouy/OY 13

(5.7)

The strain measures defined in the usual

chosen manner:

are

the

three

components

ei (i =

1,2,3)

{1}o IOl:o 1

el = exx

= gl +

_\1[g21

+ g_)

=

0

g

+

_gl T

0

e 2 =

egy

=

g4

!(g2

+

0

1

+

+ eyx

= g2 + g3 + gig3

+ g294

=

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

1

0

0

1

0

0

0

1

0

0

0

1

0

T

½gT

g+

1 0

from of the

which

expressions

tangent

for hi and

stiffness

matrix

Hi

(i = 1, 2, 3) follow.

will be described.

bl =

Begin

b2 =

0 ga

' 0 Then

from

(4.22)

we get

1 +

the

core

o si = s i + Eijej,

In full and

using

E11a_

the

abbreviations

+

0

g,

(5.s)

g,

(5.9)

(5.1o)

only the

the

derivation

vectors

1 + g4 1 + gl

g4

T + sini

(i, j = 1, 2, 3), are

'k 2E13alg3

by forming

b3 =

strains

(5.11)

"

g2

stiffness

S = Eijbib where

For brevity,

'

GL

0

1

e3 = exy

of the

the

=

PK2

al = 1 + gl,

E33g_

Ellalg2

SM

stresses

Ellg_

in the

current

configuration.

a4 = 1 + g4 we get

q- E13(ala4

SM----

(5.12)

+ Sa,

+

g2g3)

+ 2E13a4g2

-'k E33a4g3

+ E33a_

8ymm E12alg3 E12g2ga

+ +

Elaa_

E13alg_

E2:g_

+

E2ag23

+

E23a4g3

+ 2E23alga

+

Ea3alga +

El_ala4

E33ala4

+

El_a4g2

+ Ea3a_

E22a4g3

+

E13alg2

+

E_3a4g3

+

+

E23a]

E13g_ E23(ala4

+ g2g3)

Ez_a_ + 2E23a4g2

0

,S1

st

SG----

8yrnrn

sa

0

0

s3

s2

0 82

14

+ +

Ea3g_g3"]

E33a4g2 +

Ea3alg2

+ E33g_

J

(5.14)

5.3

Plate

This

is similar

motion

Bending to the

in 3D space

of membrane assumed.

previous {X, Y, Z}

stretching The

and

three

Consequently

example

in that

is allowed.

With

bending.

For

energy-contributing

nd = 3, ns = 3 and

g4 g5 , g6 three

GL

strains

are

defined

are

now

contributing

2

+

functions

+g4

arranged

as

as

+

+gig4

gradients.

Cguz/cgY

1

0

0

0

0

0

0

1

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

"0

0

0

0

0

0"

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

1

0

0

0

0

1

0

0

0

0

0

0

0

1

0

0

0

1

0

0"

0

0

0

0

1

0

0

0

0

0

0

1

0

0

T

=

O

0

lgT

g+_

T

1 gT

g+_

T

0 g2

is

COuy / OY

1 =

model

(5.15)

'0'

eyx

is capable

of six are

now

// UXJaX

0

+

plate

Oux/OY

0

exy

but

Ouz/OZ

=

0

=

the

plate

COuy /cgX

'0'

ea

thin

mathematical

gradients

0 1

freedom

a Kirchhoff

strains

'1

e, = exx = 9, +

is a flat

increased

latter

ng = 6. The

g3

structure

this

the GL

g2

The

the

+g2g5

+g3g6

=

'

1

T

' g+5"g

1 0 ,0

g,

(5.16)

g,

(5.17)

1

0

0

0

0

1

0

0

0

0

0

0

1

0

0

0

g'

(5.18) which

define

expedient For

hi and

and

example,

Hi,

i = 1, 2, 3. When

less error-prone the

following

to obtain Macsyma

one the

core

program

respectively: hl h2:

:

matrix(J1],

[0],

matrix(J0]

, [0],

[0] [0],

, [0], [0],

[0] [1],

reaches

, [0])$ [0])$

15

this

matrices forms

SM

level

of bookkeeping

through

symbolic

and

in matrices

S_

it is more

manipulation. SM and

SG,

h3: matrix([0],C1],[0],[l],C0],[0])$ g: matrix ( [gl], [g2], [g33, [_], [g5], [g6] )$ HHI :matrix(It,0,0,0,0,0], C0,1,0,0,0,0], [0,0,1,0,0,0]

,

Co,o,o,o,o,o], [o,o,o,o,o,o], [o,o,o,o,o,o])$ HH2:matrix([0,0,0,0,0,0],

[0,0,0,0,0,0],

[0,0,0,0,0,0],

Co ,o,o,i,o ,o], [o ,o,o ,o,I,o], [o,o, o,o ,o,i] )$ HH3:matrix(C0,0,0,1,0,0],

C0,0,0,0,1,0],

[0,0,0,0,0,

I],

CI,o,o,o,o,o], Co,I,o,o,o,o], Co,o,I,o,o,o] )$ bl:hl+HHl.g$ b2:h2+HH2.g$ b3:h3+iIH3.g$ SM :E1 I* b 1. transpose (b I )÷E22*b2. transpose (b2) +E33*b3. transpose ÷ El2* (b 1.transpose (b2) +b2. transpose (b 1)) + El3* (bl. transpose (b3) +b3. transpose(b1) ) + E23. (b2 .transpose (b3) +b3 .transpose(b2) )$ ratvar s (g6, g5, g4 ,g3 ,g2, gl, a5, al, E11 ,E12 ,E13, E22 ,E23 ,E33) $ SM :rats imp (SM) $ SG :rat simp (s I*HHI÷s2*HH2+s3*HH3) $

These

matrices

ments

(not

the

core

may

shown

tangent

SM(1,

be

above). stiffness

automatically

converted

That

was

this

output

reformatted

semi-automated

1) = E33g24 + 2E13(1

+ gl)g4

SM(1,2)

= E_3((1

+ gl)(1

+ gs)

SM(1,3)

= Ez3((1

+ g_)g6

+ g3g4)

to TEX

+ El1(1

+ g2g4)

by appropriate by hand

process

(b3)

Macsyma

for inclusion

here.

stateFor

yields

+ gl)_

+ E33g4(1

+ E3394gs

+ gs) + E_(1

+ E_I(1

+ g,)g_

+ gl)g3

= E 23g.2 + E3_(1+ 9_)g4+ E12(1+ gl)94 + E13(1+ g.)_ SM(1,5) = E12((1 + gl)(1 + g_)) + E:_g_(1+ g_) + E_g_g_ + E,_(1 + g,)g_ SM(1,4)

SM(1,6)

=

En(1

+ g_)g6

+ E_3g4gs

+ E33g3g4

+ E_3(1

+ g_)g3

SM(2,2) = E_(1 + g_)_ + 2E_g_(1 + g_) + El_gl SM(2,3) = E_(1 + g_)g_+ E,_(g_g_+ g3(1 + g_))+ E,_g_g_ SM(2,5) = E2_(1+ g_)_+ E_3g_(1+ g_) + E,2g_(1 + g_) + E_a_ SM(2,6)

= E_a(1

+ g_)g_

S:,,(3,3)

= E33g_

+ 2E_3g3gs

SM(3,4)

= Ea3(1

+ gl)gs

SM(3,5)

= E_3(1

+ g_)9_

SM(3,6)

= E _3g_ + E33g3gs

SM(4,4)

=

SM(4,

E

+ E_:g_gs

+ Eaag_(1

+ g_) + E_ag_g3

+ E_3g_g_

+ El__g3g_

+ E,a(1

+ E33g2g_

+ El_g3(1

+ g_) + E_3g_g3

(5.19)

+ Eng'3

+ E_2g3g6

+ gl)g_

+ E_3g_

_p_ +2E_(1 +_)g_ + E_(I +gl)_

5) = E_3((1

+ g;)(1

+ g_) + g_g4)

+ E_g_(1

+ g_) + Eaa(1

+ g,)g_

sM(4,6) = E2_((1 + g_)a_ + a_a_)+ E_g_a_ + E_(1 + g, )a_ S.(5,5) = E_:(_ + a_)2 + 2E_3g_(1+ g_) + E_a_ _M(5,6) = E2_(_+ a_)a_ + E_._(a_a_+ a_(1 + g_)) + E_a_a_ SM(6, 6) = E _gs _ + 2E_ag_g6 (which

can

be further

compacted

+ E aaga

by

introducing 16

the

auxiliary

symbols

a,

=

1 + g_ and

0

01

Y U

x

u0

1

2

:X

Xo Figure

as = 1 + gs as done

1

Kinematics

in Section

SG

=

5.3)

SGp

of 2D Timoshenko

beam

sl

0

0

Sa

0

0

0

sl

0

0

sa

0

and

=

0

0

.sI

0

0

.53

Sa

0

0

s2

0

0

0

Sa

0

0

s2

0

0

0

sa

0

0

s2

REMARK 5.1. If the plate element to which the particle doms, all additional geometric stiffness (the complementary transformation phase. Because of this, the core geometric Sop,

where

REMARK

subscript 5.2.

The

P means

element

(5.20)

belong has (as usual) rotational freegeometric stiffness) appears in the stiffness (5.20) has been relabeled as

"principal."

core stiffness

matrices

may also be used for part

of the formulation

of thin-

shell facet elements, with the proviso that global reference axes {X, 1i, Z} are to be replaced local coordinate system {X, Y, Z} with Z normal to the element midplane. 5.4

2D

Consider notational The

only

and

the

Timoshenko next PK2 mean

Beam

an isotropic

simplicity stresses shear

by a

Timoshenko

it is assumed that stress

that

contribute s2 -

axy.

plane

beam

that

the longitudinal to the The

strain

in the

axis of the energy

corresponding 17

moves are GL

(X,Y)

beam

the strains

plane.

For

is aligned

with

stress

Sl --- sxx

axial are

the

axial

X.

strain

el =-- exx

and

equations

are

and

modulus,

shear

modified The the

initial

sumptions and

section-averaged

sl = s o + Eel

from

finite

normal

the

are

position

of the

of the

Timoshenko

deformed

term

e2 --- 7xY

material.

project

described

beam,

beam may

where

E and

The

treatment

in a local

model

the

+ eyx.

G are

The

the

Young's

modulus

below

is slightly

sections

and

system

1. Under

coordinates

that

the

remain

constitutive

outlined

de la Fuente

coordinate

in Figure (plane

axis)

= exy

by Alexander,

as illustrated

centroidal

configurations

strain

s2 = s o + Ge2, of the

of a course

displacements

deformed

and

respectively,

that

to the

shear

Haugen. is attached

to

kinematic

as-

usual

plane

but

of a particle

1°

not

in the

necessarily

underformed

be written

x x0+,x0{ }0, {x+u0x}[cos0 sin0]

x = x0 + RT¢, where

u0x

tracting

and

(5.21)

uoy

are

from

X0 =

the

(5.22)

components gives

the

u_x_X= Four

displacement

gradients

n d : 2, n s -- 2 and the usual pattern:

n 9 = 4.

of the

element

contribute

g =

9 we note

g =

=

that

the

9_ g3

=

=

1) Thus

displacement

Ouy/OX Oux/OY

7 -

u0.

Sub-

"

(5.23)

for

this

gradients

case are

we have

arranged

in

(5.24)

"

can be written

Yx sin 0 - sin 0

in terms

of generalized

(5.25)

'

cos O -- 1

GL

el

vector

field

strains.

gradients

The

strains

displacement

(5.22)

'

Ouy/OY

is a generalized axial is the beam curvature.

of the

cos8

/gl// uXj0X} g2 g3

in which e = Ouox/OX strain, and _ = O0/OX form

GL

contributing

94

matrix

sin8

{ UoyUox-YsinS} + Y(cosS-

g,

For future use in Section section freedoms as

centroidal

to the

four

=

displacement

{ux}___ uy

The

,

Uoy

strain,

3' = Ouov/OX

a generalized

shear

is

1_92 gl q- _ 1 "_-g2) _ =

0 0

a T g + _g

0 18

00

1 0

0 0

0 0

0

0

0

g'

(5.26)

e2 =g2÷ga+glga+g2g4=

1

_

1

g + _g

r

0 0

0 0

1

0

0 which

define

bl

hi,

h2, H1

=

,

and

H2.

spectral

introducing

1 + g4 1-4-gl

b2 =

0 the

On

C1

the

c2 = 0

g2

core

stiffness

S r ---- EClC ---- SM

and

T W ac2c

internal

force

=

Because

beam

matrix the

elements

appears

when

subsequent

5.5

3D

The

large

transformation

As

in the

2D

G. The

necessarily

stress

the longitudinal of the left-end

free.

some

algebraic

This

description

of moving

frames, these

cross-sections A beam

4- s2H2,

(5.31)

phase.

in Section

This

term

in

9.

is largely s and

used

stiffness

is considered

a TL 3D Timoshenko

thesis

notation

geometric

extracted

is continued

in those

beam

references

from

capable a recent

with

the

DGCCF

has

been

slightly

configuration

elastic

with

Young's

of the

beam

frame

ni is attached

is straight

modulus

and

E

prismatic

to it, with

and

shear

although

nl

directed

not along

axis (the locus of cross section centroids). Axes n2 and n3 are in the plane cross section; these will be eventually aligned with the principal inertia axes

{X, Y, Z}. Initially

---- 81Hi

(5.30)

a complementary

material

is isotropically

A local reference

to simplify

X.

o 2 4- s2)H2,

article.

beam

reference

(5.29)

(5.32)

involve

following

10. The

present

the

element

v as well as Crivelli's

in Section

of the

case,

The

be written

4- s2b2.

of core equations

Felippa

phase

to fit that

modulus

Kinematics

1 _(s

SGP

transformation

Beam:

rotations.

and

the of this

of derivation

by Crivelli

edited

out

can

+ s°H2, 4-

T,

freedoms,

treatment

Timoshenko

of arbitrarily

Slbl

rotational

carrying

GCCF

last example

paper

have

vector

T -4-s°H1

T 4- Gb2b

(5.28) ,

½g2

1 0 4- _(Sl 4- sl)H1

T

SM ---- Ebib

4- SGp,

+ _g4 1-4-_gll

'

0

matrices

(5.27)

vectors

=

'

S U = ECl cT + Gc2c

S

auxiliary

g,

particle

denoted frames

of the

moving

originally

expressions.

Along

these

is schematically by

{al,a2,a3},

coincide

with

current

shown

axes

parametrized {nl,

n2, n3},

we attach

in Figure by and

2.

the We

the

coordinate further

longitudinal

displace

rigidly

system

define

a set

coordinate

attached

to the

configuration.

at (X, Y, Z)

displaces

x(X)=xo(X)+RT(X)_(Y,Z),

to _T=[0

19

Y

Z],

(5.33)

a2

Y

X, n I

Z

!!3

Figure

where

x0

describes

orthogonal position

matrix

the

position

function

vector.

The

Kinematics

2

of the

that

orients

displacement

field

sequel

3 x 3 skew-symmetric

X0(X)

The

spin

skew-symmetric

of change

of the

(a)

symbol;

=

orthogonal

cross-section,

section,

and

a3 0

a2

--al

matrix

rotation

is a 3-by-3

_ is a cross-section

are

(5.34)

(see

consistently

Figure

denoted

2).

by placing

a tilde

--a2

a

a1

=

0

a2

=

axial

T = -_RR

(_).

( 5.35 )

a3

_ is defined matrix

{el}

R with

by k = R(dRT/dx), respect

to the

T, 2O

(_O =

axial

which

longitudinal

The curvature vector is _ = axial (k). We shall also require later the orientation GO, defined as the axial vector of the skew matrix R(_RT:

/_'O = R_R

R

for example

--a3

curvature

centroidal

matrices

0

=

cross

element

is

displacement

In the

3-vector

displaced

given

is the

= x0(X)

axial

the

of the

T-I)_.

u0(X)

their

centroid

beam

u=x-X=u0+(R

where

over

-

of 3D Timoshenko

(_-'O),

variation

is the

rate

coordinate. of angular

(5.36)

All displacement the

nine

gradients

gradients

gl

=

gij appear

are partitioned

Ouy

/OX

,

in the three

=

Ouy

g2

Ouz/OX The

9-component

directly

vector

Also introduce

h_ =

With

the help

g can

be given

measures.

To maintain

of these

is gT

the

3-vectors

0 0

,

quantities,

/OY

=

g3

,

[glT

=

OUy/OZ

Oux/OZ Ouz/OZ

gT

gT],

but

this

(5.37)

,

}

symbol

is not

{0} {0}

h2 =

explicit

compactness

3-vectors:

Ouz/OY

gradient

here.

GL strain

into

1 0

,

h3

=

0

.

(5.38)

1

expressions

for the displacement

gradient

duo g2

only

nonzero

duo

=

( RT

components el

=

ell

=

e2

_

"/'12

=

-- I)h:,

of the

2e12

=

ga

GL

strain

where H is here the matrix R we find

3 × 3 identity 1

e22 = hTg2

+

hTgl

+ hTg2

The

strains el

=

+(R=-I)

eb =

+ R2a + R21R31

(5.40) =

\ dX

shear

eb, ef are

may ef,

]

1 hl + 2 dX

and

U=

/ JL

strain

that

T

+ gTHgl),

Hg3

from

the

Ra2 + R2aRa,

the

only

in a more

(5.40)

+ gTHgl), orthogonality

of the

rotation

(5.41)

and

flexural

with

nonzero

physically =

R2a = O,

strains

are

suggestive

(5.40).

form:

_'12 Jr- '_13,

e2 = 712 = 75 + _2 = aTe

+ hT_

T

e3 = 713 = 73 + Ta = hT_ b + hT_ T_" normal

torsion-induced

stored

-

'

_ ¢TK:,,

contribution energy

-

7

72, ?'3 are

LldAdX,

f 0 JA

that

u0)

a squared-curvature 9.'2-). The

+ R22Ra2

be rewritten

eb +

stretching

strains,

1 _gT-+ _( 1 ng2

2+R_a)=R=-l+½(2-2R_)=0,

confirms

el = ¢T]¢_ + ½_T_Tt¢ Here

Hgl,

+ hlTga + _(gl Note

be written

+

ea3 = 0. This

ell

can

1

matrix.

I)ha.

T

2e 3= h g, + similarly

( RT-

(5.39)

+ _g2 g2

=a_2-1+1(a_,

= R,2

=

tensor

1 T _gl

hTgl

ea = 71a = 2e13 = hTgl

and

vectors

as

g' = d--Z+ Rr _¢ = d--Z+ RrCr_' The

used

to flexure, in the

strains, shear

strains.

which

can

current

b/=_Ee1

72 and

The usually

configuration 2 1+-_ 1 G (e_ +ea_)+

o

21

73 represent last

term

bending-induced in e I represents

be neglected

(cf. Remark

is sOe1+sOe12

+sOea.

(5.43)

5.6

3D

Timoshenko

The

PK2

stresses

sxy

and

s3 =- sis

s3 = s o + Ge3.

Beam:

associated

with

= sxz. The

Core

The

spectral

1

vectors

ci = hi + _Hg

H - I. stiffness

Applying

the

the

strains

(5.40)

equations

stiffnesses

can

Sv

level

=

(:l eT,

spectral

we obtain

S U

for

--

formulas

of Section

e2 eT,

S r a form

S U _-

e3¢

similar

S U

to (5.44)

• _ =

=

0

_,-

$1

geometric

= bib1 T, $2 stiffness

sill SGp

The

=

stiffness

in

energy

(5.44)

0J

At

prestresses

internal

the

residual

s i0 , i = 1,2,3

force

vector

conjugate

=

s2bl

and

matrix

shear

stresses,

respectively.

S = SM + Sap

is obtained

ESI+G(S2+S3)

GS4

GS5

GS T

GSl

0

GS ff

0

= b2b_,

$3

(5.45)

,

= b3b_,

$4

from

(4.22).

The

(5.46)

GS1

= b2b_

and

$5

= bsb_.

The

principal

is

s2H

s3H"

s2H

0

0

s3H

0

0

contribution

is needed

9 x 9 core

S U ---- C3 cT.

the

of the

s3bl

normal

SM----

where

in terms

and

{ 161)b2+s363)

of the

The principal core tangent material stiffness is

The

s12 =

s0H]

0o

and

s2 -

s2 = s o +Ge2

for the

s°H

that

+si).

0

contribution

C2CT

except 0

= sxx,

no subscript

+|s°H [s°n T,

sll

expressed

[s°H

0

1

the

-

4.4 we obtain

GS[

have to be replaced by the midpoint stresses 7(si to ag is • = Srg + q,0 = _ + ,I_r, in which

represent

sl

sl = s ° +Eel,

be compactly

0 where

are

are

bi = hi + Hg i for i = 1, 2, 3, where

"ES + G(S[+ GS[

S U =

GL

constitutive core

i and

equations

of

(_2g)T_

[(s °+Eel)H =

|(s O+Ge2)H L ( so + Ge3)H

to the

complementary

(s °+Ge2)H

(s °+Gea)H

0 0

geometric

0 0

stiffness

(5.47)

depends

target variables in the ensuing transformation phase. Because this transformation the DGCCF, it is taken up in Section 10 after the GCCF is discussed in Section 22

on the requires 8.

6.

The

DCCF

core

TRANSFORMATION

stiffness

matrices

respectively,

pertain

is expressed

in terms

of its

model

the

in terms

of the

physical

structure

of core-to-physical

Over

in (4.19)-(4.22)

structure.

The

gradients

collected elements.

collected

the

v are

integration

scope

linear.

of the

over

Finite

displacement

g.

through

particle

To

elements

create

a combination

that

developments

used

to reduce

indexing

field

u T = (ul,

u2, u3)

the

transfor-

pertain

to an

clutter. is interpolated

as

u = Nv, where

v now

collects

N(X1,X2,X3) respect

the

element

is a matrix

to the

Xi

and

taking

the

functions

first

last

obtained

one

see

Remark

by integrating

K U = /

independent

Invariance

(4.7)-(4.8)

over

Kr

strain

element

= /voGTSrG

of v.

(DOFs)

and

Differentiating

(6.1)

(_2g = 0,

of the

the

of freedom

N = with

yields

/fg = G (_v,

4.1).

GTsuGdVo,

degrees

two v variations

g = Gv,

(for the

(6.1)

node-displacement

of shape

a

equations

domains.

by assuming

all subsequent

are

(4.13),

of each

in vector

element

CCF

and

behavior

constructed

Direct

Because

identifiers

the

finite

in vector

and

v are

element

of the

given

into

no element

an individual

vector

FREEDOMS

displacement

within

g and

element,

PHYSICAL

is subdivided

DOFs

we stay

between

individual

particles

transformations

section

mations

internal-force

to material

discrete

In this

and

TO

(6.2)

energy

reference

dVo,

variations

volume

_U

and

_2U

yields

K = /voGTSG

(6.3)

dVo,

J Vo t"

f = JvoGTO

p=

dVo,

/yoGT_

dVo,

P° = /voGT_°

dVo,

Although the dependency of S I_'et and • on g is not made implicit it must be remembered that the transformation g = Gv also appears the

ensuing

gradients Often and

algebraic are

constant

over

G is expressed xealain

For example, node

complexity,

inside in the

displacements

the

as a chain the

element

bar element while

numerical

integration

is generally

in these there.

(6.4) equations, Because of

required

unless

the

element. of transformations, integral treated

T transforms

whereas below, local

some others

are

not

and

G = T G, where

to global

23

of which

node

are

position

may

be taken

G transforms

displacements.

dependent outside. g to local

7.

7.1

The

Bar

The

core

equations

These

L0.

bar

a geometrically

now

applied

element.

two end

displacements local

for are

TL

The

TRANSFORMATION

EXAMPLES

Element

equations

prismatic

DCCF

nodes

are

coordinates

to the

The

VY1,

{X, Y, Z}

VZ1

has

at (Xl, and

)

may

TL

formulation

element

are located

(vxl,

nonlinear

bar

constant vy2,

reference (X2,

in

Section

5.1.

linear-displacement,

area

Y_, Z_),

The

vz2 ).

be interpolated

derived

of a two-node,

]I1, Z1) and

(vx2,

were

A0

and

initial

respectively.

element

length

The

displacement

node

field

in

as

uY1

fi =

_y

=

Nx

0

0

N2

0

0

N1

0

0

N2

fiz

where

N1

respect

=

1 -

to the

f(/Lo

and

reference

N2

1 G= This

transformation

5.1.

Finally, usual

transformation manner

be applied

1 0 0

to the

to the

L_ool/vo

the

0 0 -1

0 -1 0

core

shape

core

tangent

displacements

0 1

9=

--

_z which

is valid

tangent

for

stiffness

both matrix

end

nodes

in local

vectors

stiffness

(5.6)

yields

this

simple

element

is necessary.

An

in the

of a Fortran

subroutine

with

approximately

Tzx

Tzy

Tzz

giving

9i

may

efficient

160 floating

(_]T(EbbT be

point

1,2

is handled

in the

(7.4)

uz

1,2.

+ sH)l_l

in Appendix

24

=

Consequently

the

element

by

obtained

operations.

(7.3)

1_,

= Tu,

ur

J

implementation is given

in Section

{ux}

i =

is given

o]

integration form

Tyz |

= Tv,

i

derived

equation

Trr

all entries

+ sH)

, VZi),

Tyx

TT

(7.2)

and

transformation

coordinates

K = _o For

with

1Gq, Lo

matrices

(VXi,VYi

local-to-global

fly

Differentiating

0

0 1 0

Txx fl ----

(7.1)

}

functions.

dVo = _oA°GT(EbbT

(]Ts(_

to node

by writing

linear

[00

application

b2 -

are

=Ng,

we get

[--1

o7"

may

For example,

= f(/L

coordinate

_X2 { _X1 vy2 OZ2

[

(7.5)

T0

in closed of the

form

tangent

2. This

and

stiffness

implementation

no

numerical

matrix

(7.5)

forms

K

7.2

Iso-P

For the finite

Plane

case

models)

DOFs

can

As

of plane

elements

such

are the

are

are

located

{xi

=

Xi

element

Element

stress

considered

isoparametric nodal

then

in Section

ponents

Stress

freedoms

be handled 5.1,

the

denoted

reference

at

{Xi,Yi},

+ uxi,

yi

displacement

the

(i

=

1,...

may

of the

and

n)

in

(i

=

we shall with

the

that and

the

that

associated

(as usual

transformation

for

to physical

DCCF. and

{ux,uv},

be expressed

The

system

1,...

asume

n nodes,

type.

current

{x,y}

Yi + uyi},

field

purview

system,

{X,Y},

=

5.2,

models

are of translational

within

by

in Section

displacement

in-plane

displacement

respectively.

The

reference

configuration

n)

current

in the

com-

element

Co and

nodes move

configuration

to

C.

The

as VX1 vY1

, Vyn

in which

Ni

coordinates with

are

appropriate

such

respect

example,

as _ and

to X

the

isoparametric

and

physical

rI for quadrilaterals. Y,

and

tangent

all

SM

elements, the

and

SG

(7.7)

such

the

are

by

discussed

required

before

of freedom such

the

the

expression internal

2, the

relation

plates

expression has

force or drop

the •

out

interest

(4.8)

of the

second

depending

integration.

in the

Core

Gauss

For

occurs rotations

product

relation

of linear several

computer

of

implemen-

algebra

system

converted

to Fortran

or C

loop.

CCF

Congruential

Formulation g and

in elements are

exactly

52U of the of the

the

finite

with

or

GCCF

element

rotational

is

degrees freedoms,

treated. internal

quadratic

(62g) T _.

of g with 25

case

a computer quadrature

gradients

S as kernel

inner

on the

(6.3)-(6.4).

Because

in the

through

automatically

variation

stiffness in the

per

differentiation

As in the

of efficiency

and

complication if finite

tangent

as

respectively.

evaluated

displacement

shells,

appears

upon

of natural

(7.7)

GENERALIZED

between

in terms

follows

transformed

by numerical

Generalized

This

G matrix

(5.14),

encapsulated

and

core

also

and

in the

THE

written

-{- Sa)GdY,

evaluated

being

v is nonlinear.

as beams,

Recall

survive

in Section

when

/voGT(SM

or Mathematica,

8.

As

The

equations

be symbolically

Maple

functions

is

(5.13)

sparse,

may

statements

=

conveniently

matrices integrand

as Macsyma,

program

given

is most

integrand

tation

are

core

stiffness

K

where

shape

,

This target

energy form second physical

density. in 5g.

term

The may

This core either

or generalized

coordinates (the latter term is explained below) chosenin the CCF transformation phase. In the caseof the DCCF, this term drops out and S

is the

tangent

considered

core

stiffness,

so far.

But

if that

S and

Sop

are

where

-b KGp

forward

transforms two

the

---. and

spectively. the

can

be

complementary

variation core

geometric

8.1

For elements v is often intermediate and

by

of this as q.

In Section

2 it was noted

should

equations

sumed tangent 8.2

set

array

that

examines

to be independent, stiffness

matrices

Algebraic

The,Algebraic

cannot

geometric v produces

_vTKGc

a new

_V,

a

(8.3)

there

KGc

in

terms

is no "core

Instead

kernel

upon

re-

term

be expressed

to (8.2).

called of the

complementary

it appears

as a "carry

transforming.

Target

treatment

a one-shot

transformation. final

transformation

The

will be collectively

two variants in terms

the

principal

degree

of freedom

referred

target:

element

between

to as node

sets

"generalized

g and used

as

coordi-

displacements

v, is a

of choices.

variational

of generalized

as

GCCF-transformation-styles,

Consequently

the

situation

is relabeled

freedoms

T {I_ H

produces

term

be added

is the

(8.1)

and to

(_2g)

and

Generic

GCCF

process

be distinguished

development target"

as

Of course

of such

at various

can

a multistage

targets instance

KGp,

as a quadratic-form

the

identified

particular

tial,

that

stiffness

(62g)T_I_

phase

gradients.

Sac

require

replaced

tangent

--4

That

This

First,

(8.2)

term

SGp

stiffness.

Coordinates that

the

transformation

materializes

Generalized

nates"

the

geometric

that

core

(6.3).

happen.

,

DCCF-transformation

seen

stiffness"

term"

principal

as per

things

AV SGp

transforming the schematics

6g of the displacement

forward

S M

SM --* KM,

_-, symbolize

As

_

(8.1)

S G

survives

called

-k KGC,

+

term

stiffness, respectively. Second, extra term in accordance with K = KM

S M

which

S

in which

--

of the

GGCF,

of consequences

levels.

These

transformation

on the

variants

from

coordinates

qi collected

a restriction

removed

and

force

internal

qualified

are

existence examined

displacement

vectors,

q.

below.

Symbols

These K and

respectively,

and

differen-

of physical

gradients

in vector later.

as algebraic

stiffness

The

g to

ensuing

a

"generic

coordinates f are

in terms

used

are

as-

to denote

of q.

Transformation GCCF,

or AGCCF,

applies

if the

relation

get) is nonlinear but algebraic. We have g = g(q) Differentiating with respect to the q, variables yields

or

between

g (source)

in index

notation,

and

q (tar-

g_ =

g_(qj).

Og_ 6gi -

Oqj_qj

= G,j6qj,

or

6g = G_q, (8.4)

62gi

-

02gi _qj6qk OqjOqk

+ Ogi 62_j=20 _qj 26

Fijk_qj_qk,

or

62g = (F6q)

6q,

Here (F6q) is the matrix the name because

Fijk 8qk = Fkij transformation matrix.

tangent the

Enforcing

qi are

assumed

invariance

6qk; F being a cubic array. The The second term in the expansion

to be independent

of 52 U yields

the

target

tangent

+ where k --

the 1,...

yields The

entries ng.

the

are

that

with

applied the

Remark

portion and

G defined

f = [

(cf.

prestress

happens

ing the

matrix

of the

force

+

= Qi, Fkij

= Fkii(I)k

= Fkji.

geometric

vectors

= with

(8.5)

summation

Integration

stiffness

transform

+ of q

on

over

KG.

according

to the

formulas

GT_

dYo,

p=

to K U and

Kr?

/voGT_

They

dVo,

p° -

can be obtained,

/yoGT_°

somewhat

dVo.

artificially,

by construct-

equation

W is called

because the

a secant

transformation

its ng x nq entries

qj ---, 0 if 0/0

limits

must appear

in general

W

-_ G,

(8.7)

matrix.

satisfy

only

in W.)

(8.7)

,

symmetry

Generally

this matrix

ng conditions.

Using

KU=jfyoWTSuWdVo Because

in

(8.6)

g = Wq, where

V0

in (8.4):

J Vo

What

because

Kcc

transformation

+ Qii

4.1)

Q is symmetric

complementary

internal,

(6.4)

of Q

Note

variables.

stiffness

=

array G receives of 62g i vanishes

(Care

we can

has

proceed

is far from often

unique

to be given

Kr=_GTsrWdVo.

in the

secant

stiffness

to

to form

(8.8) K"

cannot

be expected

even

if S r is symmetric. REMARK 8.1. clude fixed-axis

The AGGCF is applicable to finite elements with degrees of freedoms rotations, because such rotations are integrable. Examples are provided

dimensional beams as well as plane in-plane motions are allowed.

stress

(membrane)

elements

with

drilling

that inby two-

freedoms

if only

REMARK 8.2. Why is Kcc called a geometric stiffness? Because it vanishes if the current configuration is stress free, in which case the core internal force • vanishes and so does Q. 8.3

Differential

The

Differential

(target)

is only

Transformation GCCF, available

or DGCCF,

as a non-integrable

tSgi = Gij_qj,

62gi

=

is required

OGij 6qj Oq"-"-k

or 6qk

if the

differential

relation form

between between

g (source) their

and

q

variations:

6g = G6q, =

Fijk

6qj 5qk,

27

(8.9) or

52g = (FSq)

Sq.

The transformation equation (8.5) still appliesfor tors.

But

require

no integral

a secant

g = g(q)

matrix

Q is not

necessarily

OGki/Oqj

= OGkj/Oqi.

REMARK

8.3.

REMARK constraints

8.4 Up

What are

point

the

The

linear

But

q have

the

DCCF q are

(Ogi/Oqj) metric

for

_2qj

q are

assumed the

term

applies

for the

acquires

stage

Sections

9-10

variables

rotations

are

of non-holonomic

transformations, a non-integrable

a higher

that

source

of freedom

second

of (8.4)

order

component,

v,

variables

all

survives. implicitly

next

strictly effect

defined

one.

If the

formulas

hold exists,

is that

as the

the

kernel

geo-

of

(8.10)

meaning

explicit

stiffness)

until

degrees

beterm

$2qj dVo,

final

q

congruential.

relation

net

applied

chain?

are

The

as previously

for the

previous

which

geometric are the

But

conveniently

differential

( "resolution" either

more

in a transformation

remaining

in the

illustrates

We continue in Section

5.4.

Axis

it is assumed

with

This

here

section

the

basic

A 2-NODE

here

constructed

element.

case

three-

extraction the

of its stiffness

transformation

of freedom

(and

chain thus

kernel reaches

independent),

linearly on such. It is difficult to state detailed rules that encompass all possible Instead the treatment of the 2D and 3D beam element transformations in

9.

cross

when

such

variables. are

the

in v, or only

be resolved

variables

or depend situations.

the

vec-

or equivalently

arises

because

with

force

K r, which

Furthermore

= Fkji

naturally

of freedom,

become

degrees

of a complementary

downstream

for the K v and

Fkij

DGCCF

transformations

intermediate

independent

nonlinear

cannot

form

being

of resemblance

/Vo _iGij

This

holds

be constructed.

to be independent

GCCF

in one

final

= Gij _2qj

stiffness

in the

that

degrees

points

(8.6)

Consequently

cannot

the

as nodal

have

been

variables

if the

in the

if the

(8.7),

elements

present

elements

target

happens

cause

are

finite

K whereas

exists.

Transformation

for complicated

in stages.

form

a condition

8.4. The relations (8.9) in analytical dynamics.

to this

AGGCF

of the

mechanical

rotations

Multistage

noted,

relation

symmetric;

For

dimensional finite non-integrable.

as in the

area X that

2D

the

A

-= A0

is made the

serves

two end

and

cross

six degrees

moment

section

"carrying

through

the

is doubly 28

as (8.10).

beam

element

started

The

specific

element

GCCF. and

reference

I = fA Y_ dA

symmetric

such

ELEMENT

Algebraic

centroid

terms

Timoshenko

of freedom,

of inertia the

forward"

BEAM

of a 2D, isotropic to illustrate

nodes,

to pass

for

TIMOSHENKO

derivation

example

has

techniques

so that so that

are

fAY

dA

length

constant

L0.

The

along

the

= 0. Furthermore

fA y3 dA

= O.

The

element

linear

displacement

shape

field,

defined

by uox(X),

uoy(X)

and

O(X),

is interpolated

with

functions: e

troy Uox

N1

0

0

N2

0

0

0

N1

0

0

N2

0

0

0

N1

0

0

N2

}

0

Vyl l)xl

[

O1 Yy2 Ux2

= Nv,

(9.1)

J

, 82 where

N,

= 1 -

OaOX

eare

YX2

over

the

set constant

over

terms

of q it is convenient

of these

51

and

each

Consequently

naturally

-- vX1

-

00

Lo

'

Stress

Resultants

coordinates

we take

cross

section

when

obtaining

to integrate

in terms

generalized

vx2

OX

of generalized

coordinates.

appear

7-

Consequently

_-

02

OX

-

7

_;

-- 01

Lo

'

(9.2)

element.

tation

ties

'

Coordinates

As intermediate are

N2 = 1 - Na = X/Lo. OUoy

Lo

Generalized

quantities

and

-- UXa

OX

constant

9.1

(X/Lo)

over

of cross

coordinates

the

and

may

beam

=

vectors

0].

These

as cross-section

four orien-

matrices

and

internal

section.

The

resulting

quanti-

below.

In terms

cross

stress-resultants

auxiliary

[e

be viewed

stiffness

the

section

qT

as shown

bi listed

in (5.28)

forces

in

become

sin0 } /::1} {+Ycos0){3}{ 7 - Yx sin 0 0

=

0

b2 =

'

1 + g4 l+gl

0

cos 0 l+e-YxcosO

=

g2

7 -

'

Yg sin 0

(9.3) The

well

known

stress

resultants

force V and bending moment the beam cross section:

N=

of beam M.

They

theory

are

A sldA=EA(e+7(e

are

obtained

the

force

by integrating

2 + 72 ))+

1

axial

1EI_2

N, the

transverse PK2

shear

stresses

over

+ N O,

0

v = fA s2dA

= GA,w._

(9.4)

+ V °,

0

M

=/a

slYdA

= -EIxaJ_

+ M °,

0

where

w_

generalized

=

(1 + e)cos0

skew

strains.

+ 7sin0 In (9.4)

and

w.y =

N °, V ° and 29

7cos0

-

(1 + e)sin0

M ° denote

initial-stress

can

be

resultants

viewed (stress

as

resultants in Co, also called prestressforces), which

tt is the

usual

doubly-symmetric fAo y3 dA

linear

correction

cross-section

has

In addition

shear

been

to N,

theory,

omitted V and

appears

the

in the

C

]A

assumption, from

M,

A -- A0, I = fao y2 dA, and A, = /IA, in of Timoshenko beam theory. Because of the

factor

the

expression

following

residual

slY2

da

a term

= EI((e

the

third-section-moment

for M.

higher

force

containing

order

and

moment,

tangent

which

is absent

from

the

stiffness:

+ ½(e 2 + 7_))

+ ½ET-lx _) + C °,

(9.5)

-

(9.6)

0

in which

"H = fAo y4 dA.

If terms

C where this

r = V/_ relation

is the

is only

C O = (g

radius

exact

of gyration

Transformation

The

differential

from

(5.25)

= (N

of the

and

cross

vanishes

2,

section.

approximate

fA s_YdA

g°)r

If such

terms

are

retained

otherwise. identically.

This

serves

as a check

of the

Matrices relations

required

to establish

the

tangent

transformation

are

obtained

as

0g

1

-Y

5g = _qq 6q =

0 0 0

0 -Ycos 0

Oq 5q=

The

that

neglected,

- N°)(I/A)

if r 2 = 7-l/I

REMARK 9.1. One may verify strain distribution equations. 9.2

in _2 are

1

_-Sv

= L-'_

transformation

G = G1G2

=

i

0

-1

0 -1 0

0 0 0

relating

Ysin01{ }

sin0

-Y_ cos 0 ] -cos0 ] - sin 0

1

0

0

0

0

1

0

-1 0 L0 - X

0

0

1

0

0

5g = G 5v

may

67 6x

J

(9.7)

= Gx 5%

50

5vy1 = G_ 5v, { Svxl

(9.8)

}

X

be obtained

as the

product

-1

0

Y(cosO+(Lo-X)_sinO)

1

0

Y(-cosO+X_sinO)]

1

0

-1

Y(sinO-(L0-X)ncosO)

0

1

Y(-

L0

0

0

-(L0-X)cos0

0

0

-Xcos0

0

0

-(L0-X)sin0

0

0

-Xsin0

sin O - Xn

cos O)

J

(9.9) but

it is more

mentation) Observe second

instructive

to perform that one

transformation

the

first

(from

(as the

transformation

q to v) but

the

well as conducive

transformation

is linear.

second

one

to higher

phase (from

g to q) is nonlinear

Consequently can

efficiency

be done 3O

in the

computer

imple-

in two stages.

we have simply

to use

through

and the the

algebraic

whereas

the

AGCCF

for the

first

DCCF.

9.3 The

Internal internal

(5.32)

Force force

for @ and

Vector

vector the

in terms

matrix

G1

fq = /Ao GT @ dAo

=

of q, denoted given

in (9.7):

f_ f_

=

by fq, is obtained

N7

application

of (9.8)

and

integration

over

the

-

core expression

-fq

=

(9.10)

"

Vw,

element

length

1

f = fo L° GT2 fq dX

the

- -Mw, M_ sin +0 C_ + V cos 0 -Mtcw.y

Finally,

from

yields

q

+fq 7Lof_

(9.11)

•

+ 1Lof$ This

vector

9.4

Tangent

Transforming ness

satisfies

translational

Stiffness

Matrix

to generalized

(GT(SM

JA

entries

of K_t, K_(1,

q produces

three

components

of the

tangent

stiff-

obtained

through

1) = EA(1 ) = EA(1

K_t(1,3

) = EI_

+ Q) dA

+ g_p

symbolic

manipulation,

+ K_c.

(9.12)

are

sin 2 0 + EI_ 2 cos: 0,

+ e)7-

sin0cos0

GA,

+ EI_ _ sin0cos0,

((1 + e)(1 + cos 2 0) + 7 sin 0cos 0),

= El_2w-r

cos 0 + GA,_z_ sin 0,

) = EAT 2 + GA,

K_t(2,3

) = EI_

cos 2 0 + EI_

sin 2 0,

((1 + e) sin 0cos 0 + 3'(1 + sin' 0)),

4) = EI_2w_

sin 0 - GA_w,

= EIw, _ + ET-la 2,

I(],(3,4)

= EI,¢ ((1 + e)7(cos 2 0-sin

K_a(4,4)

= EIx_'¢2g + GA._z, 2. stiffness,

which

(9.13)

cos 0,

K_(3,3)

geometric

= K_t

+ e) 2 + GA,

K_t(2,2

K]_(2,

principal

+ SG)G1

o

g_t(1,2

K_(1,4)

The

coordinates

matrix:

Kq = f

The

equilibrium.

2 0) + (_2 -(1

is readily 31

worked

out

+ e)2) sin 0cos0),

by hand,

is

-M

=

The

new

K_c.

term

Its

contributed

source

(5.25) The

and

(9.3),

entries

matrix

where

the

-M_cos0-Vsin0 (9.14)

0

AGCCF

to K q is the

Q introduced

complementary

in Section

of g and

8.2.

cI, = slbl

geometric

The

entries

+ s_b2

may

stiffness

of Q are be obtained

Qi.i

=

from

respectively.

of Q were

symbolically

generated

by the

[eps_, gamma_

,kappa_,

,gamma,kappa,theta_

phi={l,

following

Mathematica

module:

thet a_, Era_ ,Gm_, Y_] :=

[{g, hl, h2 ,Hl, H2, el, e2, s i, s2 ,bl ,b2 ,phi,

q={eps

V cos 0

Cn 2

components

OmatrixOf2DTimoBeamElement Module

sin 0 -

C

0

by the

is the

(02gk/OqiOqj)_k,

Mx

-Msin0

N N symm

cos 8

i ,j ,k},

I, i, I};

g={eps-Y*kappa*Cos [theta] ,gamma-Y*kappa_3in[theta], -Sin [thet a] ,Cos [thet a] - I} ; gg={{g hl:{{l}

[[I]] } ,{g [ [2]] },{g [_]] ,{0} ,{0} ,{0}} ; h2:{{O}

},{g [[4]] }}; ,{I} ,{I}, {0}};

HI={{I,O,O,O},{O,I,O,O},{O,O,O,O},{O,O,O,O}}; H2={{O,O,I,O},{O,O,O,I},{I,O,O,O},{O,I,O,O}}; el=(Transpose

[hi].

gg+

(1/2)

e2=

[h2]

. gg+

(1/2)

(Transpose

sl:Simplify[Em*el]

[thet

gamma-Y,kappa*Sin

For [i=l,

{4},

i++,

For

Return

[ [1,1]

gg)

] ;

[ [1,1]

] ;

; [thet

a], O, O} ; ,

[theta]

a] } ;

[thet

=Q [[i,j]]

. H2.

, l+eps-Y*kappa*Cos

[sl*bl+s2*b2] {4}] ;

i