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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 ,
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"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
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-. Truss #lement
$em'ers o! the trusses are su';ected to a%ial !orces only, 'ut their orientation in the plane may 'e at any angle to
the coordinate directions selected. "igure 3.3 shows a typical case in a plane truss. "igure 3.@ (a) shows a typical
mem'er o! the truss with
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"ig. 3.@
(ii) Unit displacement in coordinate direction -G
This case is shown in "ig. 3.@ (c). In this case a%ial de!ormation is 1 % sin 8 and the !orces de&eloped at
each end #ire as shown in the !igure. EA . n
P E Fsin 6 L
i . EA .51- E P cos EBBBBBBBsin cos
H . B EA . 2 HHk 12 E P sm E Fsin 8
EAk ,, E —P cos EBBBBBBBBBBsin cos3- L
KAk d2 - —P sinO EBBBBBBBBBBsin
- 8@- L
(iii) Unit displacement in coordinate direction 3,
#%tension along the a%is is 1 % sin and hence the !orces de&eloped are as shown in the "ig. 3.@ (d) EA n
P E Fcos
L
EAh-i = —P cos EBBBBBBBBBcos- 8
I N
)
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EA,, E —P sin EBBBBBBBBBcos sin
-3 L1 -n . FA 2 n
E P costJ EBBBBBBBcos 833 L
i . . EA . K ABi E P sin0 EBBBBBBBBBBBBBBBBBcos smd
L
(&i) Due to unit displacement in coordinate direction @,
#%tension o! the 'ar is eual to l% sin, and hence the !orces de&eloped are as shown in "ig. 3.@ (e). EA . n
P E Fsin 6 L
EA . n n-P costJ E suitJ costJ
L
EA
h,A = —P sin EBBBBBBBBsin- 8-@ L
7 . EA .%3@ E P cos8 E Fsind cose
7 . . EA . - k AA = P sind E
Fsm 8 L
The sti!!ness matri% is
.(3.K)
Lhere > and ! are the direction cosines o! the mem'er i.e. > E cos 8 and ! = cos (M B 8) E sin 8 " (&)
I@
L Bcos- 8-#$% 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
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: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
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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)
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EA
L
##
(a)
"ig. 3.H
1 113B(')
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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
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...(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
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"rom the last euation, and then,
n