Finite Element Analysis by S.S. Bhavikatti, New Age Int. (2005)

8/20/2019 Finite Element Analysis by S.S. Bhavikatti, New Age Int. (2005)

1/11

8

=

3

Matrix Displacement Formulation

3.1 INTRODUCTION

Though mathematicians, physicists and stress analysts wored independently in the !ield o! "#$, it is the matri%

displacement !ormulation o! the stress analysts which lead to !ast de&elopment o! "#$. In!act till the word "#$

'ecame popular, stress analyst wored in this !ield in the name o! matri% displacement method. In matri%

displacement method sti!!ness matri% o! an element is assem'led 'y direct approach while in "#$ though direct

sti!!ness matri% may 'e treated as an approach !or assem'ling element properties (sti!!ness matri% as !ar as stress

analysis is concerned), it is the energy approached which has re&olutioni*ed entire "#$.

+ence in this chapter, a 'rie! e%planation o! matri% displacement method is presented and solution

techniues !or simultaneous euations are discussed 'rie!ly.

3.- $TRI/ DI02C#$#NT #UTION0

The standard !orm o! matri% displacement euation is,

456 {8} = {F}

where 46 is sti!!ness matri%

789 is displacement &ector and

{F\ is !orce &ector in the coordinate directions

The element k.. o! sti!!ness matri% may'e de!ined as the !orce at coordinate i due to unit displacement in

coordinate direction j.The direct method o! assem'ling sti!!ness matri% !or !ew standard cases is 'rie!ly gi&en in this article.

1. :ar #lement

Common pro'lems in this category are the 'ars and columns with &arying cross section su';ected to a%ial !orces

as shown in "ig. 3.1."or such 'ar with cross section ,


2/11

8

=

"ig. 3.1(c)

"ig. 3.-

EA ? — o£

If 5 = 1, P =

:y gi&ing unit displacement in coordinate direction 1, the

!orces de&elopment in the coordinate direction 1 and - can 'e !ound ("ig. 3.- (')). +ence !rom the de!inition o!

sti!!ness matri%,

EA , EA — and 5-i=~ —

0imilarly gi&ing unit displacement in coordinate direction - (re!er "ig. 3.- (c)), we get

(a)

4

T L,

()

1

55

A, E

L _

! "1~*

L

(a)

P I-----► N----- P

I 1 I

()

EA

L


3/11

8

=

-. Truss #lement

$em'ers o! the trusses are su';ected to a%ial !orces only, 'ut their orientation in the plane may 'e at any angle to

the coordinate directions selected. "igure 3.3 shows a typical case in a plane truss. "igure 3.@ (a) shows a typical

mem'er o! the truss with


4/11

8

=

"ig. 3.@

(ii) Unit displacement in coordinate direction -G

This case is shown in "ig. 3.@ (c). In this case a%ial de!ormation is 1 % sin 8 and the !orces de&eloped at

each end #ire as shown in the !igure. EA . n

P E Fsin 6 L

i . EA .51- E P cos EBBBBBBBsin cos

H . B EA . 2 HHk 12 E P sm E Fsin 8

EAk ,, E —P cos EBBBBBBBBBBsin cos3- L

KAk d2 - —P sinO EBBBBBBBBBBsin

- 8@- L

(iii) Unit displacement in coordinate direction 3,

#%tension along the a%is is 1 % sin and hence the !orces de&eloped are as shown in the "ig. 3.@ (d) EA n

P E Fcos

L

EAh-i = —P cos EBBBBBBBBBcos- 8

I N

)


5/11

EA,, E —P sin EBBBBBBBBBcos sin

-3 L1 -n . FA 2 n

E P costJ EBBBBBBBcos 833 L

i . . EA . K ABi E P sin0 EBBBBBBBBBBBBBBBBBcos smd

L

(&i) Due to unit displacement in coordinate direction @,

#%tension o! the 'ar is eual to l% sin, and hence the !orces de&eloped are as shown in "ig. 3.@ (e). EA . n

P E Fsin 6 L

EA . n n-P costJ E suitJ costJ

L

EA

h,A = —P sin EBBBBBBBBsin- 8-@ L

7 . EA .%3@ E P cos8 E Fsind cose

7 . . EA . - k AA = P sind E

Fsm 8 L

The sti!!ness matri% is

.(3.K)

Lhere > and ! are the direction cosines o! the mem'er i.e. > E cos 8 and ! = cos (M B 8) E sin 8 " (&)

I@

L Bcos- 8-#$% 8 sin cos- cos 8

Bcos 8 sinK

sin- 8 #$% 8 sinK

sin- 8

P>- &!-' 2

- &

_ EA &

!2-

~ L -' 2 — ' - &

— -!2 & !

EA

cos- 8 #$% 8 sin Bcos- 8 cos8 sin8 sin-

8 Bcos8 sin8

Bcos 8 sin

Bsin- 8


6/11

:eam #lement

In the analysis o! continuous 'eams normally a%ial de!ormation is negligi'le (small de!lection theory)

and hence only two unnowns may 'e taen at each end o! a element ("ig. 3.8). Typical element and

the coordinates o! displacements selected are shown in "ig. 3.8 ('). The end !orces


7/11

de&eloped due to unit displacement in all the !our coordinate directions are shown in "ig. 3.K (a, ', c, d).

"ig. 3.K

"rom the de!inition o! sti!!ness matri% and looing at positi&e senses indicated, we can write (a) Due to

unit displacement in coordinate direction 1,

(d) Due to unit displacement in coordinate direction @,

I! a%ial de!ormations in the 'eam elements are to 'e considered as in case o! columns o! !rames, etc. ("ig. 3.H),

it may 'e o'ser&ed that a%ial !orce do not a!!ect &alues o! 'ending moment and shear !orce and &ice &ersa is also

true. +ence sti!!ness matri% !or the element shown in "ig. 3.Q is o'tained 'y com'ining the sti!!ness matrices o!

'ar element and 'eam element and arranging in proper locations. "or this case

EA

L

##

y 1 3 8 H M(a)

1 3

"ig. 3.8

1-#H AE& ) 6

E&

1- E' A K E' 2 L) (a) L2 K E' () L

L2

1- E' 6E& 2E& K E'

1} L L2

. ...................... ! A $l + A.

6E&

2 . .11-H

y BK E'

AE& &

, ? 12E& , _6E& , ? 12E& k u-—- 531?

(') Due to unit displacement in coordinate direction-,

, _6E& G ? AE& 6E& S1- ~ - S-- B — S3- B -

(c) Due to unit displacement in coordinate direction

3,

, ? 1-H , ? 6E& , ? 1-HS13

? S-3

? S33

? ~y~ L L L

k A/ -

k 02 ~

K>H

&

2E&

L

_ 6 E'

L2

, 2E&

%!4 E &F A-@ L

P 1" & ' &

r , F& & 0L2 '& 21

LB '1"'&

L1"

'& L

& 21 '& 01

A3@ VV6E&

A@@ 0E&

...(3.H)

E A

2E& 6

1- K

1} L2 1} L

6E 0

6E 2

L- L L2 L

E A

2 6

2E 6

L2 L2 L2 L

6 2

6E 0

L2 L L2 L

...(3.Q)


8/11

EA

L

##

(a)

"ig. 3.H

1 113B(')


9/11

The !ollowing special !eatures o! matri% displacement euations are worth notingW

(i) The matri% is ha&ing diagonal dominance and is positi&e de!inite. +ence in the solution process there is

no need to rearrange the euations to get diagonal dominance.(ii) The matri% is symmetric. It is o'&ious !rom $a%well=s reciprocal theorem. +ence only upper or lower

triangular elements may 'e !ormed and others o'tained using symmetry.

(iii) The matri% is ha&ing 'anded nature i.e. the non*ero elements o! sti!!ness matri% are concentrated near

the diagonal o! the matri%. The elements away !rom the diagonal are *ero. Considera'le sa&ing is

e!!ected in storage reuirement o! sti!!ness matri% in the memory o! computers 'y a&oiding storage o!

*ero &alues o! sti!!ness matrices. The 'anded nature o! matri% is shown in "ig. 3.M.

"ig. 3.M

In this case instead o! storing / si*e matri% only % 3 si*e matri% can 'e stored.

3.3 0O2UTION O" $TRI/ DI02C#$#NT #UTION0

The matri% displacement euations are linear simultaneous euations. These euations can 'e sol&ed using

Xaussian elimination method. 2et the euations to 'e sol&ed 'e


10/11

...(3.-)

i.e. 4A5{} = 7!t9

The Xauss elimination method consists in reducing matri% to upper triangular matri% and then !inding the

&aria'les n , % ..., ...., 2 , %, 'y 'ac su'stitution

7( 1: To eliminate %, in the lower euationsW

(i) "irst euation is maintained as it is

(ii) "or euations 'elow 1,

and bjp E !t, B F ft

t the end o! this, the euations will 'e

The a'o&e process is called pi&otal operation onan. "or pi&otal operation on ;, no changes are made in 5 row

'ut !or the rows 'elow

>,(5B1) t

and


11/11

"rom the last euation, and then,

n

Finite Element Analysis by S.S. Bhavikatti, New Age Int. (2005)

Documents

Transcript of Finite Element Analysis by S.S. Bhavikatti, New Age Int. (2005)