p. • r

aFor the system shown:

1. Write the member force-displacement relationships in global coordinates.

2. Assemble the global stiffness equations. 3. Show that the global stiffness equations contain rigid-body-motion

terms. E = 200,000 MPa. c A = 18 x 1<P mm2

10.928 mI. r r

4m

1

_ "Q,::olo. ,

I J .'m<!>""'~~'f' 0'" 'rr\ 'rt/C> '')or

0b {l:t.. = 3

= 7 7. I T/ l.

~/o/uJ r .. T

101'" • 'f>

bN l.,.,.m.

2 r r,"1,

C'a~ C<a:" "'.1'L

r:..A.~ = '1 - /I .'(\

o~

~"~ F. .~

It

a:> (!) F"~

I 0·5 - o· 0

'ih>~ , (}.s 0· ­I:"" r Fa..,

r~b .. l

,

!c ~ ~ 200 J2x /0 3

:: 32 . 3 kfJ/.".,.",£L )k

10. 9:2 P Y It) 3

® (£) @ <!) Etc-

r Fsk. r ' F: 1><- Aj/. 000- 0 - I· 0 ([) ­

I @)31. '1, 4-3 0 ":. '='F./' < C! ...= F:6t. 10"," 0 eDb,fi t 1:.10 CD

- 375.

rt

0·150 ~. "'33 - 0·75

0- so - . t...:: 37 . 0

~. r.L~()QJ.~•

:. ... / .A'l. " /1.:. ,K"

k dbKIJ '!: I. ~,

~

~ :to'=" 'I 0" + ~7.5·oo

,kCU, , ac:. f:'..t ~ /1 '1 ~I =10.':/. I' (- Q. S..... " 7 ·0

k ..L 7-0,:/. II (-.,. !:"'K'3 ~ 13 ~

1~'1f ~ rl'i ':/t>9·/ (0. r 0 "3. S ...

::, • (q / . I

oc. .:: 0(- 0'''1 ~ - ::l J. ,

/('5 k,s 37"­.. ":: ':. 37 ~. C . .,~ 16'J - ~?+< " t"

I. IOC

k31 - I ,, -t ~:JI '= '1 ~. /l (-0'$ CI ;- ~75.of) (o·~ ~ ­

I Q~/(~l ~ :n. of ftl.

~)

c ~ 707· 1/ (0'5 ) 375.0.0 O.:l SO ::­

J ': k'3fA" :: /e''1. II ( o· So.. :'

~ -: :: ";h.1·" (- o·Sh: :.

k,s A,.s ::- 375. -0' ~3 :.

')'7 C~ I) ,,, 5''.)J. ~ &~J. = co ­-

Ai &-:. K3a - k'33 k 3 ~ 70 9-1/ . s .. 0+- :s) 9· ~ 3 (, eo)- =

,,,~ b_~e -t t;).. ':" ;;.r,9· , (- o· Sa'" =

6.t. ~3S b]S 3::'1· 'H (-/.B,."

:: -:19·l.I~ Cokll ~-

, 7'r ) ·s )+- 3:lCJ· (If't =- ~ .. -=

,rk 3~ 9. 't3 (., oil

Itt. ... 0- R~ '"

or K!s : ~sr -J9.1+· I. -I =-'75. o( .? So)-=

Q

~'1.4 ( ),

s:r. :: ks,

of k(t 0 .... 3'1<;k" c.

Global stiffness equations in matrix form

6.348 -1.912 -3.536 3.536 -2.812 -1.624 ~l IP1

4.473 3.536 -3.536 -1.624 -0.938 ~2P2

6.830 -3.536 -3.294 0 ~3P3 =1OZ • Sym. 3.536 0 0 ~4P4 .. 6.107 1.624 ~5•P5

0.938 ~6P6

3. Rigid body motion. Adding rows 1 and 3 of the global stiffness matrix yields the vector:

L2.812 1.624 3.294 0 -6.107 -1.624J

which is the negative of row 5. Therefore, there is linear depen­dence, the determinant is zero, and the matrix is singular. This is a signal that, under an arbitrary load, the displacements are in· definite; that is, there may be rigid body motion.

Iq /./ g or ( ~ .,......., ...... ";""1.. J/': ::1

c:4f1t'j·3 .. r .,.3 3·.5

~ 3 3· sss .r

..,- '6~ .375

.,- 93'1

.,Gi2.'1'iS

-)S3·Sss

-3 CJ·~:) ./

~

3 ::l · 5

;:> 10. 6~ '" -: 16'.315 V'

.,.9,3·7