De elementenmethode, toegepast op een ringschijf, die diametraal, door twee radiaal-gerichte puntkrachten wordt belast

De elementenmethode, toegepast op een ringschijf, die

diametraal, door twee radiaal-gerichte puntkrachten wordt

belast

Citation for published version (APA):

Brekelmans, W. A. M. (1969). De elementenmethode, toegepast op een ringschijf, die diametraal, door twee radiaal-gerichte puntkrachten wordt belast. (DCT rapporten; Vol. 1969.019). Technische Hogeschool Eindhoven.

Document status and date: Gepubliceerd: 01/01/1969

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hìgoì 05062828 Brekelmans begin comment prog,nr.

05062828,

Ir. W.A.M. Brekehans. Toepassing van de elementernethode op een ring, die diametraal door twee puntkrachten wordt belast. Gebruik gemaakt werd van een trapeziumvomnig plaatelement. Er zijn, bij aanwezigheid van N elementen, 4N onbekende verplaatsingsgrootheden: displacement

-

method, N >

-

3 Afmetingen van de ring: buitendiameter: 2R2 binnendiameter: C!R1 RI/(R2

-

RI) s dikte :h Materiaalconstanten zijn G en nu. De grootte van de krachten is P, naar buiten gericht positief. Aangrijpingspunt kan worden gekozen: voor

x

= O: buitenrand bij Ii =

-

+

pi/2 voor

x

= 1: binnewand bi3 fi =

-

+

pil2 (per definitie) (per definitie) Volgorde van invoergegevens: N, s, nu,, x; integer N; 1V := read; PNIMTTEXT( kaantal gebruikte elementen, N:$); MBFIXT( 2, O, 11); NLCR; IWR; begin integer i, j, k, m, x; M

+

4-2,

1:(4

% N

+

4)1,

a[1:8],

Q,[1:8,

1:81,

T[1:8,

1:8],

u[1:(4 X N i- 4)l; IUI real eps, s, nu, nul, pi, ksi, f3k, ck, tk, fi, sfi, cfi, tfi, r, epsr, epst, gart, sigr, sigt, taurt; array

XI.

boolean interchange; integer

C1:(4

x rJ)I; I^__ real procedure inprod(i, il, 12, ai, bi, e);

212

il, 12, c; i, il, 12;

-

real ai, bi, e; i := i1 1 uti1 i2

Gd.

cc: :y c + ai X bi; _u1I. inprod := c _I__ end inprod; __y real procedure CKQUTD8CUMPOSITION( n,

A,

LU, p, interchange, eps, singular);

-

value n, interchange, eps; integer n; real eps; __I array

A,

LU; begin integer i,

J,

k, kkj real g,

r,

s; array;

nom~:n];

p; Boolean I interchange;

kal.

singular; for k :=

1

step 1 until n

&

begin s := O; _y end coGutation of rownorms; _s P

-

for j := 1

e

1 until n I- do be& LUCk, j] := r :’= A[k, j]; s :E s

+

abs(r)

---

end; if s = O __L then be@$=] := k;

a

singular -__.*_ end else norm[kl :E s

q := 1; for k := 1 1 until n Gin s :=t O; for i := k

1

until n do := r := inpgd(j,

1,

k

-

1,

-

LU[i, j], LU[j, k1, LU[i, k]); if i = k V interchange then %gin r :" abs(r)/nomCirif Y"

>

s then begin s := r; kk := i end end -1 _I_

-

_u_ end search for pivot; if s

<

eps then begin p[n] := k; &o singular end; if s

<

g then g := s; zk] := kk; _I _u_ -L_ if kk k then ggin nommj := nom[kl; _c. for j := 1 1 until n do begin r :=

,

j~~k,~] ilrii*I end interchange rows; r := LU[k, k]; := LUCkk, := r end

-

for j := k c

1

step

1

untiln begin LU[j, k] := LU[j'xTr; Luck, j1 := inprod(i, 1, k

-

1,

-

Lurk, i], LUrí, j], Luck, j])

-

end k; CROUTDECOMWSITIOEJ := q

-

-

end CXOUTDXCOMPChSITIOM;

L-

rocedure CROUTINVEXGE(n, LU, pI X ); n; tinteger n; array LU, X; integer array p; inte er i, j, k, kk; real r; array save [l :n];

XL$%I

:= î/Lü[n,

nl~

for k begin __I := n-î

steJ?-

1 until î

do

for i := k i- 1

a

1

until n do for i := k c

1

îìiñGIndo

%,

k] := hnp kn n,X[ m :=

-

LULI, kl; saveCj1, 0); for- j := k r := Luck, for j := k

XE,

jI := I_ end inverse of

L

_^__ i-

1

steE

1

until n do savekjl :=

-

LÚík, j]; kl; S_e 1 until n inprod(i, k + 1, n, saveci], X[i, jl, if j 3 k then 1 else O)/r x

u;

__I_ .U IQ-

-

for k := n ste

-

1

until begin kk

:=$I;

ifr$ begin r := XE, k] j end permutations _I __I end

CWTINVXRSKSE;

1

do k then __L- for XCi, k] :=

1

ste

1

until n UdL i,

"kks

: =

II

. .

m n

. .

. I ti * .

s

1 1 rn e.

o

c V 1 11 t -14 II .1 . 1 r.7 ' r3

E

U N x

8 E

mi

+

-

4 "

rnx

a03

co

4

rn

TL1,

11

:= TL2, 21 := TC3, 51 := T14, 61 := T[5, 31 := T[6, 41 := T[7, 71 := T18, ,81 := ck; T[1, 31 := T[2, 41 := T[7,

51

:= T[8, 61 := sk; T[3, 71 := T[4, 81 := T[5,

11

:= Tc6, 21 :E -sk; comment bepaling van de matrix K, hier: Q[1:8,

1

:O]; A-

@)

for i :=

1

ste begin Q[i, ?=

QIm1

:= O;

1

until 8 do for j := i step

1

until 8 do __I- 1- _I s-

--

_.___ for k :=

1

ste

1

until 8 do for m :=

1

ste

1

until 8 do

m,

jl

:=

-p!

Q j,

i3

--

:= Q\.i,lf]TTCk, i]

--f

X

2

k, zn-T[m)j]; end; _II -11_1 coment bepalirq van de eoefficientemtrix ZC

1

: (4N

+

4),

1

: (4N

+

4)l; for i := 1

1

until (4 X N + 4) do for j := i

1

until (4 X N

+

4) do &[i, j] := Zij, i] := O; -_DI --w. 1_ II

-

for m := O ste u4 x m

+

-i;9(4 x m

+

j)~ :=z

xn4

x m + j),

&

m

TTJT

:=

1

until (N I-

1)

do for i :=

1

ste

1

until 8 do for j := i ste

1

until 8 do __I -_I -__L L_s m

+

i)* X m

+

j)]

i--Q[i,

jl;

comment de nu aangegeven rijen en. kolommen moeten uit de bepaalde matrix Z worden verwijderd: 3, 4, 43

+

3,

4N

+

4, We houden over: %Cl :4N,

1

:kN]; for i := 3 stea;2

1

until (4 3< N) do for J := i P€UNTTE312'(@lementen van de mtrix Z[

1

:4N,

I :4D]$); NUR; NEW;

1

until (4 x N) do X[i, j] := Zij, i] := Zli +

2,

j + 21; UI_ __II

-

_I I__ for m := 1 step

1

until ((4 9< N) : IO) do Gin PRINTTEX'T(~j~~FIXT(e,-O,

((G

-

1

) X

10)

+ 1 )); PRINTTEXT(ktm$); ABSFIXT(2, O, (m 4( IO)); NLCR; _L_ for i :=

1

1

until (4 x N) do NLCR; PRïNTTwfi=$); klBsEXT(2, O, i); SPACE(4); for j := ((M- '8) X 10 i- 1) stz 1 until

(mx

10) do begin FUJT(3, 1, %[i, jl); SsCE(2)7-

--

_II end; end; NLCR; IVLCR; NXR;

-

_I_ end; PRINTTmT(kj=$); ABCFIXT(2, O, (((4 X N)

-

: 10) X

10

+

1));

PKLNTTEXT(+t&); ABSFIXT(2, O, (4 x N)); NUR; __u

end; NLCR; _L_

Q

o

:-z i NUR; PRLNTTmTfi=$); AESEXT( 2, O, i); SPACE(4); for j :=

(((4

X N) : 10) X 10 + 1) begin FZOT(3, 1, Z[T,

J1

1;

SPACE(2); end; i Lmtil

(4

X

N)

do

1

until

(4

x NI

2%

_Lc_ u*___ _I_ NUR; NLCR; Matrix Z wordt nu geinverteerd; CRûUmZCaMa?oSITION(

(4

x

a),

Z, li;, p, interchange, eps, singular); CROUTINVENSE((4 x N), Z, p, X);; _._.I_ comment Berekening van de dimensielaze verplaatsingen u in radiale richting =- uGh/p

-

v in tangentiale richting = vGh/p die dg volgende structuur heeft: 21, ~2~

21

u en worden ondergebracht in een vector uil :4N

+

41,

22,

23

.

.

.

. .

.

v2N

+

i,

22N

+

2;

-

x := read; PRINTTE.XT(b= _I if

x

= 1

-

then PRINTTEXT( NUR; NE~UR; MER; AESFIXT(1, O, x); SPACE(4); kracht aan de bimeman else PRINTTEXT( kracht aan de for i := 1 for i :=

5

n4XN+ := ‘mX N i-

411

:= 2 u do u[il :=

x

X 0,s X Zii,

(4

‘36 N

-

I)]

i- (1

-

x) X 0.5 A

Zei,

(4

x

a)];

i731

:= urll :=

o;

1 until

(4

X

N

+ 2) do

dil

:= x X 0.5 9( Z[(i

-

2),

(4

X N

-

I)] + (1

-

x) X 0.5 X &[(i

-

2),

(4

x N PWLNTTEXT(@e dimensieloze verplaatsingen: u in radiale richting = uGh/p en NUR; NUR: v in tangentiale richting = vGh/p$); for i := 1 1 until (N

+

1) do begin k := 2 X i- i; m := 2 x i;

-

P _. PRINTTEXT

+u$);

ABSFïXT(L1, 0, k); E’LDT(4, 1, u[4 X i

-

31);

SPACE(4); 1, u[4

x

i

-

11); SPACE(4);

1,

u[4 x i,

-

21); SFACE(4); 1, uC4 x i]); NUK; NUR; PRINTTEXT( v

)T

ABSFIXT(2, O, k); ?RINTTEXT( v ); AESFIXT(2, O, m); PRINTTEXT({i); f&SFIXT(S, O, m); end; I__

e

E)

Conment Het doel van de nu volgende berekeningen is de bepaling van de spanningen in diverse punten. Biervoor is nod: We bepalen nu de matrix Q(

1

:8,

1

:8), die deae parameters geeft uit de verplaatsingen van een element, die reeds bekend zijn; dat het verplaatsingsveld volledig bekend is voor e& element, mrarw4 at1

1

tm at81 I_ for i :=

1

ste begin Q[i, ?= Or end;

1

until 8

g;

j :* i step i --I until 8 q_ for k :=

1

step

1

I___ until. 8

&

Qci, j] := Qci,

jl

e Zfi,

kl

X Ttk,

jl;

-

coment Voor elk element worden. de spanningen bepaald in een concreet aantal punten, In grenspunten vinden we dus meerdere uitkomsten voor de spanningstoestand; I_ for m :=

1

%

1

until N begin NLCR; NUR; NLCH; PRINTTEXT( -$nummer

..

-

i ;= 1 step i, until 8

dcq

begin aCil :i- O; for j := 1 st;e~ i YII- until 8

&

aTi1

-

end; L_ element :+) := aLi] + _; ABFIXT(2, O, m); QCi,

jl

x d(4 x m NLCR; -44- NUR; _I for i :=

-1

steg

1

until

1

begin fi := i X ksi; sfi := i

><

sk; cfi := cos(fi); tTi := sfilcfi; TAB; l?RINTWT(~i=$); FXXT(2,

2,

((i

+

2

X m

-

1)

X ksi X 180/gi)); SPACE(2); PNINTTI"TT( kratled); MLCR;

-

for j := O

9

1

until 2

&

r := s i- 0.5 X j; TAB; SPACE(

2);

PRIMTTEXT( kr=r/(

-

RC?sRl)=$); ABSFIXcE(

2,

2,

r); M;LCR; gart :=

-2

x aC21 x sfi x cfi c (af31

+

a161)

x

(cfi x cfi

-

si?. x sfi) c a[4J/(r

x

cri) i 2 )G a171

x

sfî x cfi

+

aC81 x tfi/(r

x

cfi);

I

1

a

I

‘ I

_.

I I f I

r

I Y o

t

I

t

I 1

i

53

I

I I I I O _o L s , O O '6 I I

k

O

a

$

3

3 O a ci \> i I m

\

f

y

I

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L J 'I I

'\

I

o i h

I I I I I

I

h

a

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O O

i .

.

.- X I I . Y X

I

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X X k X X I x

\

I

X t X x

\

\ X

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u c

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\

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x4

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t . r

a

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x 4 X X

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