Minimum Potential Energy
-
Upload
ahmad-sultan -
Category
Documents
-
view
7 -
download
1
description
Transcript of Minimum Potential Energy
1. Part BIWO-03 Energy Methods, FEM Class of 2014
Topic Minimum of potential energYDue Date:
2011112014
Access
Technische Universitdt DresdenDepartment of Civil Engineering
ComPulsory
Formulate an approximate solution for the computation of the
moment under consideration of large deformations (2nd order
ls,t
nl\li\
))
lnstitute for Structural AnalysisUniv.-Prof, Dr.-lng. habil. M. Kaliske
1) maximull!:glqfLgttieory) and without
consideration of the axial strain energy'
.so - 50 kN
po - 10 kN/mx
p(x) - po t
Use the ansatz-function u,y(x) - az' x2 +9' x3 .
Compare your solution to the solution derived in the seminar'
Z) For the depicted thin plate, please compute the approximate solution for the dis-
placement w(x,y) due to the vertical load p(x,y) - po.
E - 2.707 kN /m2t - 0.72m
v-0.2po - 10 kN /m2
^ L*= 3m
Use the given ansatz function
(x2\ (y' y \w(x,y) - " \e) \E- ,,)Show that the ansatz function fulfils the boundary conditions'
E - 2.L07 kN /m2
L-lm
Is - 1.}-a ma
/ x3\/(x) - ,'\r- *)
Technische Universitfit DresdenDepartment of Oivil Engineering
lnstitute for Structural AnalysisUniv.-Prof. Dr.-lng. habil. M' Kaliske
Op.tional
3) Complete the computation of the truss introduced in the 2nd serninar. Use the poten-
tial of internal energy derived there and build the potential of external energy due to
the lsads Fy and Fp. Compute the nodal displacements'
tF,+ E -2.tav kNlnf
A - $.Otmz
L --1m
Fv-50kN
FH - too kN
lF"+
Probl"* t" lMr Sryci*c s&*t ;ffi,&a .
v**o g- * F"yro
7 "c*=Ztg =a
EX1 "o
o s h*ar gtTt". * n*tJilu tP..
Fr. = €.€-xrl i{,t.ar$o..l. ?-1€**:ff*-\ t
T4= 5 *, JIftr€x{tii= ) Ee.Jttd
r.li. \* ei e q
Now , AIw-"t Snhli*
fl"ir;t"J- N, . , 4ol/ 3'd:'ll
( t Ju o *,J gti=r*,1
( ru" tt*.t t!,rr., *t 1gwi"J( lvo sha*,i*1 ** t'.gfJ.J D,re -)
lda , d ?rr<r,
o sbr''.i,rn - dxpl*"e".et* "t\*#*r*skP *
.., d"t7 ! -a4
d,X
f,;r
d).( tr* l€z-1
It i " f e 5 ti Jv4v
* -tt6 : {4.-1+1*o;
: &{!f* g"-f*
Q t"*ar*l *ft*i* rq-.^fi(f: ar --'
'r(r (*+, *l1r,*
34r h
I- l't
dn
[r* 6 e*n r*-fl. ok**r*o#*os
. ot,r*tnt Q:4 e
'd4
4^=*_--=d*
V {'"- @
a 1q"-q.t) = &i,^.ot-+"g*d'* ' o
++*'=t*. ^dr** q,J eG"i,E)
tn=# * =-u"'&
fi =* tl ( 'v"6)' dv = -i t; :*'r i Jv
frb"dfr =rz +
?rz ) aA
tua g*,y
F++. - ,tu*= "[n* ( t*
'-tct' \ \ cf'-;- --)i=dx
z =
;1(:r)'-
dx
S, i
o *erqtJ
fl**-J* €pr7 )
1-""dLlL
-2
?(
1'" TL
-jo
. ?ta#*J 4- exkr,*[ Ul,tl
dd lro*
{
,r14 oru- d,'rfV"b,'lVl l"*r
W, -[q"or-r]c*)&x - F',.o tx,L)
od,r *fury i
to-''q. 6 &Je.
- otr - Jz. *tf
-d.4={t (r-(t-4| *'g'd*u(*)= - ) -.
*r ( xlx'f+**'Fi,,ffi
Lt* (w" L) '
r qvrs.fz *r^dfi--^
laorJ*tr4 c.rn.rlr'Tior'r$ :)
1f (*=o ) . o
*(t-,t-) = y,p)x
d (il-.o) = oJeri.,ilives J o**.^lQ
'Wfiut= 2azcc ttftt) - 2e(.2-
r<;r**=f k -d1)
z(Z*. + €as x) dr
7
A.t
(
o-V
ok
-f *6.f.^.
L
I prut -u..d
?--f?-/;J6rJ-Ti-
= frtEZ
Jo
,-zE. Toi f , uL * zeo? *L + 2h a,zcr rx) At - *itu.,.. ccj + 3e ": ;:,';;:
LL
? | (",.f * a.j'tL\)drc - i \(, i.+tnlalx\+tLa,a3ll f )J*
'+Lra:n<ge a?+.2hqz.*34i # b^:4'3( ";4 +?t'o'o3!
+ L^. * r ag+l: - * [u,: + *7 al ++r2e2 ""+]:
+T,?
-L tt
!- T2c,) 1)'<*)'Jx- -
:
lt'<*l' - S*z
(ar rrt + a4 zt3 ) dx
L
'.I)x)J'x- -
Lrx)iz z=;rr"!{,-*)
L? a-27 s 3a'g x? )
,(
ll=
- -z\s os x_)+':::)
f=
ar = Erolro: L+ 6 ,? f* 6 q.qs tJ -*[^, f * G q!
7
(o,+ +q3 $l i (toi
2
,.sL+2
?"r"rf J
f3
-?V
tn a.l Lt-- ;3arorL')
TL = zEr.Lfi + 6 Er" t*, 6 tt,t*?,r- trr,t{- } Er }4-?ur"!.
-1,?,o-t? a"- t ?-f *. ',,-tSrf * Zr-r?;-==- ,'=r1)'lA./
-:,)' -
- $LT"L'..,
Q Ero Le. + 6 EL Ltor';E',t^rf - \ ?" L= _q3
s- f o. LLo= : C)
l.S- L'a, = o
JO
z
^! rLL'nzf t - ,t4
,eI5
? nr.fo. - 1 s. Lh o" Zs- LB qr * 1 E[o l] o, 7,I
L'1r*
t
f_L-' :
5,
L=1 rr,
6 e xt-rerna.r.na cor^JiG-.n o*J. &Ae^r,^.,^-fiorn 4 r.rn krno,-)rrs'.
d'll = ., +7
do.
4=o -YJo.
tAEI,Jrr* $ e tuL?ar-z Et, tlo,
-f! E["Lo. -!- s- .=z-3 - j s- Lh q,z
.+ ELf o,
l-, siwycli {;,r}t*n- u)€ s,bsf' l-'^k il. ff*7*"/iP'cs '
2 x tc>a wNl*'le KM n , lo KN/n
H'+
Za6o@#.
3eLr + SBaB aa a 2.5
a525 ctz r ITqt'O Qs*-2
:==t 0.a = a.6ao x lo-traz= -2.a73 x lo-l"t
. De1Er.*^**'* *L fi.r, ef
t4cr) -- - Er{il . tt" {x)
'l'"c*) =.2a2 + 6a3x e
irr*t= {t 4) p'h.X-= o
l,lnor = - 2 r tra x,o' r (,i'Tz
I P1*, . t/ltr
( o.ze - tv "s3s "L"V-o +' tly"(*)
'l-ro )
) , , o't''
= '[9 3Z r( to-k
=62 T1cnl = to''
xro-qf= ; {";78 kN ^wr
I n +t" taconJ S**,*r l4** "
3-d l€ - t-U t,Zr"t-lr* J,'44*^*
L'tnet+ tt"f-dpreci*z lce caulle .De ircreo.se"l, tt* nu
^ bcr
unka.tJns r
- t. hb2 kt-m
- 3i '/.
3,o1€
1k3 r..s,J,{- N = z a?€ l<//' *, ISIt- i5
LUore
?* b{** g r
#Pm,$) r . - = ls y<W[*z f*a-.3ra L= *"t?r ra*
E . 2 r ,lo? kN/,a' Ly,Zw T* o-2
" t.t"fl' t^;cot'u ) = a' t*l G, h)
k.:1:aj:t5.::.**r(re .n,0). O ,g$.
Wt.x-rg*o). O gg
,u ( x,A.r) : c: *,".1X{?t-'o,&} =c) *
dol,*-,k,.^r "o0-_o^*J. -fl",*t.^ ,
a*(*,y)= eT G L)*)(*,j)=u# (+ ilu)o*,u.,)).-a- b L+ -Z) = + ({ -5)
, xt r z -'t q. ?-ur)(*,3)=nElq) =E x-
u**cx,&}= 6L ft*{k d *?r.(S-r)
. ?.ta,dr*t *+ i^la""J E,*r19.'
T; = +Il(u*, *;),))' -z (r - tr)(w*,*oa- '.:i)] J"&J-
.?,kt;J + <*ta.djtpb-'^ --Fsv{-*€*
fr '- [ fc*,3) k) ( *,b) J,r"J#-At
, iJzrJ prfaJ.J,
rt; =+ \^[": i* ,Lu**dry*ot))- 'Z(r't]) ( Q^t't'ur)- -"3{A*
1q = + U, rz T7 +T3 -z(t-?s ) (Ir-In)l
(e (+ 3r'r*q3L (/ ")q- ai l"\ ,,
a-L2+ ts
hzl, =: q' +os
r,= \ orrrz) r*J6-= I* t(4 ?|) J,cdpA3
=#\x,r*i t4 -b)rs- hr+:rl t .p {1".
=* ?))(+-+) *, T, = b o,
a4 =l
-{ *b')*LL= ) d^*
fr7
I J,"r)"
2' 17+ € ).
r== [ *&,r*4& =](fr 4 *-U
l,=L i xqJ,,' 3Zl4 o
.- oL / j/'
fs=,+ (?ltIr=Au*io^+,
3Z
=* t+f {4-8" *#}.'
4"# (*) ($-a.z)
# 1; , + f r, +zTz *Tu -z tu:r]l{ L -il,)}
B -- ! ,1,-t- ,=,, - ? 'iql, ry e,'l't = 3c'ec) kN-orffi ur) - t2 (t u.z')
fi* rsoo t * o'-h* **; - t.€ {-**'t67so LtL[, €
a
.-*, \4t: Isr:iLl^ I JI+
5 (+ * r *-,f';* *tg
-bd)]
6 e yF, ,-l 7-iar'-ft"/ '
=-toq (4)(9' 3L; / \ stf
.-to,,_ (+) ( +'ii:-2 Lr-L,a-c.- = 'o,t'e '1 J - 5
rr ' +f 0G Coiaplet? Do-l€rtl I o.J I-'r t
_+^tu
v*, li; r E i67 sa27
-#,r@ m*n t n1,l'z'eJ i o
^- [) t"t nCt /€:
G = \ f ( r,t) u; ( *, b) ol'r .,\h
-iod-t*Jx-L', Lx
- to o. r =:l:- tt 3L;
LL?'t(L.1 \
L=Lf
-L )luLy/ d-
g'):'ZL?
L; ")7ry/
2) = l0o,-z-/
d*
ou **o
Lx
9a't--4 - 3350e: e"6Z? xto-e
L96
ali-27
Qo) +'€ =o167so
d.*
o?r,g', =*(+)( + +)