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Season of mists and mellow

fruitfulness,

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Close bosom-friend of thematuring sun;

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Conspiring with him how to load andbless

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With fruit the vines that round the thatch-eves run;

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To bend with apples the moss'd cottage-

trees,

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And ll all fruit with ripeness to the core;

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To swell the gourd, and plump thehazel shells

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With a sweet kernel; to set buddingmore,

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And still more, later owers for thebees,

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ntil the! think warm da!s will ne"ercease,

F S h ' b i 'd th i

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For Summer has o'er-brimm'd theirclammy cells

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Who hath not seen thee oft amid th!store#

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Sometimes whoe"er seeks abroadma! $nd

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Thee sitting careless on a granar!oor,

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Th! hair soft-lifted b! the winnowingwind

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%r on a half-reap&d furrow soundasleep,

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rowsed with the fume of poppies, while th!hook

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Spares the ne(t swath and all itstwin)d owers*

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And sometimes like a gleaner thoudost keep

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Stead! th! laden head across abrook;

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- "patient loo#"

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Where are the songs of Spring# A!, whereare the!#

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Think not of them, thou hast th! musictoo,

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While barr)d clouds bloom the soft-d!ing da!

A d t h th t bbl l i ith

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And touch the stubble-plains withros! hue;

en n a wa u c o r e sma gna s

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en n a wa u c o r e sma gna smourn

mong e r ver sa ows

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mong e r ver-sa ows"borne aloft

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r sin#ing as the light windives or dies;

sinking

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And full-grown lambs loud bleat from hill!bourn;

d i k t i d ith t bl

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edge-crickets sing; and now with trebloft

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The redbreast whistles from a garden-croft;

An gat er ng swa ows tw tter n t e

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An gat er ng swa ows tw tter n t eskies+

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Trusses

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At element le"el, Ue4 We Ue4 "irtual strain energ! of internal

stresses

We4 "irtual work of e(ternal forces

acting through "irtual displacements

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We now assume a simple displacementfunction to de$ne the displacement ofe"er! material point in the element+

suall! use low order pol!nomials ere

u 4 a5 6 a7x

uis a(ial displacement a5, a7are constants to be determined

xis local coordinate along member

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The constants are found b! imposingthe known nodal displacements ui, uj

at nodes iandj

ui = a5 + a7xi

uj = a5 + a7xj

ui, ujare nodal displacements xi, (jare nodal coordinates

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lettingxi4 8,xj4 L, we get

a54 ui

a74 9uj-ui):L

We can write

d? 4 nodal displacements

u x

L

x

L

u

u d

i

j

=

=1 [ ]{ }N

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Strain is gi"en b!

where @= is a matri( relating strainto nodal displacement 9matri( of

deri"ati"es of shape function

[ ] [ ]

du d[N]{d} [B]{d}

dx dx

1B 1 1L

= = =

=

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d?-.

Stress and strain are constant in amember

e$ne internal "irtual strain energ!for a set of "irtual displacements>d? to be T

e ! " # d =

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- 4 "irtual strain

- 4 stress le"el at e3uilibrium

dV4 "olume

0irtual work of nodal forces is

We4 >d?T>f?

Then, "irtual work is gi"en b!

( ) { } { }T T

d$ d % =

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Substituting and rearranging gi"es

Banceling >d?Tgi"es k=>d?4>F?where

/or thermal problem

( ) ( )T T

o

T T T T T

o

[B]{ d} &[B]{d} & d { d} {% }

{ d} [B] &[B]{d}d { d} [B] & d$ { d} {% }

=

= +

[ ] [ ] [ ]' B B= T

V

E dV

{ } { } o1

( % &)1

= +

oT =

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for a truss we get

this formulation method also appliesto 7-d and C-d elements

[ ]' =

EA

L

1 1

1 1

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cedure for irect Sti1ness Dethod 9isplacement Deth

1. *is+retize into %inite elements, denti%y nodes, elements

and num-er them in order.

2. *e$elop element sti%%ness matri+es [e] %or all the

elements.

/. )ssem-le element sti%%ness matri+es to 0et the 0lo-al

sti%%ness matrix "[] [e]#. The size o% o% 0lo-al

sti%%ness matrix total d.o.% o% the stru+ture in+ludin0 at

-oundary nodes. )ssem-ly is done -y mat+hin0 elementdispla+ement 3ith 0lo-al displa+ements. )lso de$elop

appropriate %or+e $e+tor "-y addin0 element %or+e $e+tors#

su+h that e4uation o% the type [] {u}{(} is o-tained.

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5. )pply 'inemati+ -oundary +onditions. 6ithout applyin0

-oundary +onditions, [] 3ill -e sin0ular. "minimum

num-er o% -oundary +onditions re4uired is to arrest 78i0id

Body9 displa+ements#.

:. ;ol$e %or un'no3n displa+ements {u} " {u} []n+e displa+ements are determined %ind

"a# rea+tions -y pi+'in0 up appropriate ro3s %rom the

e4uation {(}[] {u}, "-# (ind element %or+es {%}[e]

{ue}, "+# &lement stresses 0i$en -y {e} [*][B]{ue}.

2rocedure for irect Sti1ness Dethod

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1 2 3

F1, u1F2, u2 F3, u3

Boundary Conditionsu1=0, u2=0

7A E .

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7A, E, . A, E, .

[ ]2 ?2)&

?2 2@1 ?1L

?1 1

=

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>

Feactions

{ } [ ]

{ } [ ]

1

/

A)& PL 2P

( 2 ?2 A 1L /)& /

AA

)& PL P( A ?1 1 1

L /)& /A

= =

= =

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.lement /orces

A, E, .f5

f7

7A, E, .

A, E, .

1 1

2 2

21

/2

&lement 1

% u1 ?12)&

% ?1 1 uL

1 ?1 A 2p /2)& PL

?1 1 1 2p /L /)&

&lement 2

u% 1 ?1)&

u% ?1 1L

1 ?1)&

?1L

=

=

=

1 p /PL

1 A p //)&

=

2P

/

2P

/

P

/

P

/

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u5 u7 u7 uC

[ ] [ ]1 2

&lement 1 &lement 2

1 1 1 1)& )&

1 1 1 1L L

= =

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1 1

1 1

2 2

5 22 1

2 2

2 2

/ /

5/ 2

;tress in element 1

u u1 1& &B &

u uL L

u u 1.: A & 2.A 1A 2AAN mmL 1:A

;tress in element 2

u u1 1& &B &u uL L

u u 1.2 1.: & 2.A 1A

L 1:A

= = =

= =

= = =

= = 25ANmm

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irect .lement /ormulation

truss element acts like 5-d spring l GG trans"erse dimensions

pinned connection to other members

9onl! a(ial loading+ usuall! constant cross section and

modulus of elasticit!

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A4 cross section area

E4 modulus of elasticit!

L4 length

k AE

L=

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Assume displacements are muchsmaller than o"erall geometr! "ertical displacements of horizontal

member produce no "ertical force Sti1ness matri( is written in local

element coordinates aligned along

element a(is want sti1ness matri( for arbitrar!

orientation

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rotate coordinate s!stems usingrotation matri( F=

displacement components in global

coordinates are related todisplacement components in localcoordinates b! >d?4F=>d? >d? 4 displacement in global

coordinates >d&? 4 displacement in local element

coordinates

j

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L

AE

x

y

i j

ui ui

vi vj

pi

Pj

ij

st

'

A1 +olumn

'

A

=

rd

'

A/ +olumn

'

A

=

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th

AA

5 +olumnA

A

=

nd

A

A2 +olumn

A

A

=

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start with member onxa(is, elemente3uations are

or >k&?>d&?4>f&?

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pi uix

xy

y

pj uj

vj

i

vi

j

i i i i i i

i i i i i i

at node i

u D u +os" # $ sin" # p D p +os" # 4 sin" #

$ D u sin" # $ +os" # 4 D 4 sin" # 4 +os" #

= + = +

= + = +

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At node i

A similar matri( can be obtained at node H

i i

i i

u D u+os sin

$ D sin +os $

=

i i

i i

p D p+os sin

4 D sin +os 4

=

i i

i i

C C

C

u D u+os sin A A

$ D $sin +os A A

u D uA A +os sin A A sin +os$ D

=

C$

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Datri( F= is*

+os sin

sin +os+os sin

sin +os

A A

A AA A

A A

A A

A AA A

A A

=

c s

s c

c s

s c

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Similarl! , force components arerelated b! >f? 4 F=>f?

Eocal force displacement relation is

k&=>d&? 4 >f&? global force displacement relation is

k=F=>d? 4 F=>f?

using fact that F=-5 4 F=T, we getF=Tk=F=>d? 4 >f?

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then k= 4 sti1ness matri( in globalcoordinates is F=Tk&=F=

2 2

2 2

2 2

2 2

+ +s + +s

+s s +s s['] '

+ +s + +s

+s s +s s

=

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Structure e3uation is k= >? 4 >/?

k= 4 structure sti1ness matri(

>? 4 nodal displacement "ector >/? 4 applied load "ector

D Di C i i i

i

i

i

i

C

C

*B{u u } note u D u +os" # $ sin" #

u {+ s}

$

u

$+ s A A?1 1 &

uA A + sL L

$

= = +

=

{ }

i

i

C

C

u

$&+ ?s + s

uL

$

=

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/inite .lement Dodel

usuall! use e(isting codes to sol"eproblems

user responsible for creating the model

e(ecuting the program

interpreting the results

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arrangement of nodes and elementsis known as the mesh

plan to make the mesh model the

structure as accuratel! as possible

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for a truss each member is modeled as 5 truss

element

truss members or elements areconnected at nodes

node connections beha"e like pin Hoints

truss element beha"es in e(actagreement with assumptions

no need to di"ide a member into morethan 5 element

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such subdi"ision will cause e(ecution tofail due to zero sti1ness against lateral force at

the node connection where 7 members arein a(ial alignment

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there is geometric s!mmetr! often possible to reduce the size of

problem b! using s!mmetr!

need loading s!mmetr! as well

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/ig+ C-I and C-J show s!mmetricloads and the reduced model need to impose e(tra conditions along

the line of s!mmetr! displacement constraints* nodes along the

line of s!mmetr! must alwa!s mo"e alongthat line

changed loads* the load at the line ofs!mmetr! is split in two

B t i t i t

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Bomputer input assistance

a preprocessor is used to assist userinput

re3uired inputs are

data to locate nodes in space de$nition of elements b! node numbers

t!pe of anal!sis to be done

material properties displacement conditions

applied loads

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interacti"e preprocessors are preferable !ou can see each node as it is created

elements are displa!ed as the! are created

s!mbols are gi"en for displacement andload conditions

usuall! allow mesh generation b! replicationor interpolation of an e(isting mesh

allow inserting nodes along lines allow entering a grid b! minimum and

ma(imum positions plus a grid spacing

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truss element consists of 7 nodenumbers that connect to formelement

other information for truss is modulus of elasticit!

cross sectional area

data can form a material table assign element data b! reference to

the table

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boundar! or displacement conditionsare set b! selecting a node andsetting its displacement

do not o"er constrain a structure b!prescribing zero displacementswhere there is no ph!sical support

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loading conditions are set b!selecting nodes and specif!ing forceor moment components

check model carefull! at this point

A l i St

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Anal!sis Step

mostl! transparent to user small truss models ha"e enough

accurac! and performance for an

accurate solution a large model has a large number of

elements and nodes

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numerical solution ma! not beaccurate if there are full matrices

get better accurac! if the nonzero

terms are close to the diagonal reduces the number of operations and

round o1 error 9banded matri(

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in /. model, element or nodenumbering can a1ect bandwidth good numbering pattern can minimize

bandwidth di1erent methods based on node or

element numbering

to minimize, plan numbering pattern sonodes that connect through an elementha"e their e3uations assembled closetogether

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Kn /ig+ C-L, node numbers areconsidered, M&s show nonzero terms

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Kn /ig+ C-N, node numbers areconsidered

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man! programs ha"e bandwidth orwa"efront minimizers a"ailable

most programs will keep original

numbering for displa! but use theminimized number scheme

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