1. Part BIWO-03 Energy Methods, FEM Class of 2014

Topic Minimum of potential energYDue Date:

2011112014

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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', =*(+)( + +)