TE´MA 4: Technicka´ teorie skorˇepin (1)fast10.vsb.cz/brozovsky/data/nlm/p2.pdfza´kladnı´...

TEMA 4:Technicka teorie skorepin (1)

• tenke skorepiny – viz predpoklady Kirchhoffovy teorie pro desky

• vnitrnı sıly typicke pro steny i desky (vzajemne se ovlivnujıcı)

• ohybovy stav: mx, my, mxy, qx, qy

• membranovy stav: nx, ny, nxy

1

Technicka teorie skorepin (2)Ohybovy stav:

y

x

ττ

σσ

τ

τ

qxy

yx y

x

r

z

yq

x

xz

y

yx yz

xy

mm

mm

x

2

Technicka teorie skorepin (3)Membranovy stav:

n

r

zy

x

n

y

x

yx

xynn

3

Technicka teorie skorepin (4)Merne sıly v membranovem stavu:

Nx = σx h [N

m]

Ny = σy h [N

m]

Nxy = τxy h [N

m]

x

yx

xyn

z

n

n

n

y

τxy

yx

h

τ

σ

σx

y

y

��

��

x

� ��

� ��

� ��

� � ��

� � ��

4

Rotacnı symetrie (1)

• Predpoklad: membranovy stav napjatosti• Podmınky: rotacne symetricke zatızenı, pode-

prenı nesmı rusit membranovou napjatost

o

t

R R

5

Rotacnı symetrie (2)Vnitrnı sıly nx, ny:

nxxr

ny

o

z

y

zx

αr

Smykova slozka nxy = 0 (podmınky symetrie).

6


r

r pxzp

αϕ

ϕ

x

d

(r +dr)ddαr dx

ϕ

nx

nx+ xdn

ny

d

yd

dαα

n

n

y

∑

Fi,x = 0:(nx +

dnxdα)(r + dr)dϕ− ncrdϕ− nyrxdαdϕ cosα + pxrdϕrxdα = = 0

7


r

r pxzp

αϕ

ϕ

x

d

(r +dr)ddαr dx

ϕ

nx

nx+ xdn

ny

d

yd

dαα

n

n

y

∑

Fi,y = 0:nxr dϕdα + nyrxdαdϕ sinα + pzrdϕrxdα = 0

8

Rotacnı symetrie (zjednod.) (5)

r

r pxzp

αϕ

ϕ

x

d

(r +dr)ddαr dx

ϕ

nx

nx+ xdn

ny

d

yd

dαα

n

n

y

dnxrdα

− nyry cosα + pxr rx = 0nxrx+

ny

ry+ pz = 0, kde ry =

rsinα

9

Rotacnı symetrie (6)Zjednodusenı pro sılu (nebo vyslednici zatızenı) Q ve vrcholu:

t

R R

QPodmınka

∑

Fi.y = 0:

2πrnx sinα +Q = 0

Sıly:

nx = −Q

2πr sinα

ny =Q

2πr sin2α−

pzr

sinα

10

Kulova ban (1)

fa

b

q a =f2 + b2

2f

Q = qπr2 = qπa2 sin2α

pz = q cos2α

nx = −qπa2 sin2 a

2πa sin2α= −1

2qa

ny =qπa2 sin2 a

2πα sin2 α=

(

1

2− sin2α

)

qa

11

Kulova ban (2)

b

a f

q

α

Q = q2πa2(1− cosα)

pz = q cosα

nx = = −1

1 + cosαqa

ny = =(

1

1 + cosα− cosα

)

qa

12

Kuzelova ban (1)q

f

ar

rx =∞

Q = q2πa2

pz = q cos2α

nx = = −1

2 sinαqa

ny = = −q a cosα cotα

13

Kuzelova ban (2)

f

ar

q

rx =∞

Q =qπ a

cosα

pz = q cosα

nx = = −1

2 sin 2αqa

ny = = −q a cotα

14

Ohybova teorie:podmınky rovnovahy (1)Podmınky rovnovahy:

∑

Fi,x = 0 :(

nx +dnx

dxdx

)

adϕ− nxadϕ + pxadϕdx = 0

∑

Fi,y = 0 :(

qx +dqx

dxdx

)

adϕ− qxadϕ + nydϕdx + pzadϕdx = 0

∑

Mi,y = 0 :(

mx +dmx

dxdx

)

adϕ−mxadϕ + qxadϕdx = 0

15

Ohybova teorie:podmınky rovnovahy (2)Po uprave:

dnx

dx+ px = 0

dqx

dx+

ny

a= 0

dmx

dx− qx = 0

16

Ohybova teorie:geometricke rovnice

εx =du

dx

εy = −w

a

17

Ohybova teorie:fyzikalnı rovnice (1)

nx =E h

1− ν2(εx + νεy) =

E h

1− ν2

(

du

dx− ν

w

a

)

ny =E h

1− ν2(εy + νεx) =

E h

1− ν2

(

−w

a+ ν

du

dx

)

18

Ohybova teorie:fyzikalnı rovnice (2)Pro nx = 0 (protoze dnx

dx= −px, tj. nx zavisı jen na px):

ny = −Eh

aw

Momenty:

mx = −Dd2w

dx2

my = νmx

19

Ohybova teorie:zakladnı rovnice ulohy

Dd4w

dx4+

Eh

a2w = pz

Rovnici je mozno resit stejne jako rovnici steny nebo desky.

20

Aplikace: kruhovy valec (1)

Dd4w

dx4+

Eh

a2w = pz =⇒ D

d4w

dx4+4

c4w =

pz

D,

kde

c =1

4

√

3(1−mu)

√ah

Pro ν = 0, 2 (beton): c = 0, 768√ah

21

Aplikace: kruhovy valec (2)

Dd4w

dx4+4

c4w =

pz

D,

Partikularnı resenı:

wo =a2pz

Eh

Obecne resenı:

w1 = C1f1 + C2f2 + C3f3 + C4f4

22

Aplikace: kruhovy valec (3)Funkce fi:

f1 = e−xccos

x

c

f2 = e−xcsin

x

c

f3 = e−l−xc cos

l − x

c

f4 = e−l−xc cos

l − x

c

Konstanty Ci se urcı z okrajovych podmınek.

23

Aplikace: kruhovy valec (4)Okrajove podmınky:

• volny (nezatızeny) okraj: mx = 0, qx = 0

• kloub: mx = 0, w = 0

• vetknutı: w = 0, dwdx= 0

24

Aplikace: kruhovy valec (5)Srovnanı membranoveho (M) a ohyboveho (O) resenı pro zatızenı hyd-rostatickym tlakem:

L

aLγ

MEMB. OHYB.

xq� � ��

� � ��

nγ xmγ

� ��

n

� � ��

� � ��

� ��

Viz Teply, Smirak: Pruznost a plasticita II, str. 97 – 105

25

TEMA 5:Nelinearnı mechanika

• typy nelinearit:

– konstrukcnı

– geometricka

– fyzikalnı

• metody resenı nelinearnıch uloh.

26

Typy nelinearit

• konstrukcnı nelinearita – napr. jednostranne vazby nebo prvky puso-bıcı jen v tahu (jen v tlaku),

• fyzikalnı nelinearita – vlastnosti materialu nejsou linearnı pruzne (neli-nearnı pruznost, plasticita, lomova mechanika,. . . ),

• geometricka nelinearita – respektujeme vliv deformovaneho tvaru kon-strukce na vnitrnı sıly, muzeme uvazovat navıc take velka posunutı,pootocenı.

27

Konstrukcnı nelinearitaF

F

• jednostranne vazby – vazba pusobı jen v ur-citych situacıch (napr. v tlaku),

• vyuzıva se mj. resenı kontaktnıch uloh (vespojenı s Winklerovym nebo jinym modelempodlozı),

• vyzaduje iteracnı resenı.

28

Jednostranne vazby (1)

29

Jednostranne vazby (2)Model:

podlozka

uFEM 0.2.46

Time: 1CS: CART

23. 09. 2008

x

y

z

30

Jednostranne vazby (3)Reakce (klasicke vazby):

podlozka

uFEM 0.2.46

Set: 1: 1.000Results

23. 09. 2008

-7.135765e+02

-6.243795e+02

-5.351824e+02

-4.459853e+02

-3.567883e+02

-2.675912e+02

-1.783941e+02

-8.919706e+01

0.000000e+00

5.488281e+01

1.097656e+02

1.646484e+02

2.195312e+02

2.744141e+02

3.292969e+02

3.841797e+02

4.390625e+02

x

y

z

31

Jednostranne vazby (4)Deformovany tvar (klasicke vazby):

podlozka

uFEM 0.2.46

Set: 1: 1.000Result: s_1

23. 09. 2008

-7.135765e+02

-6.243795e+02

-5.351824e+02

-4.459853e+02

-3.567883e+02

-2.675912e+02

-1.783941e+02

-8.919706e+01

0.000000e+00

5.488281e+01

1.097656e+02

1.646484e+02

2.195312e+02

2.744141e+02

3.292969e+02

3.841797e+02

4.390625e+02

x

y

z

32

Jednostranne vazby (5)Napetı σx (klasicke vazby):

podlozka

uFEM 0.2.46

Set: 1: 1.000Result: s_x

23. 09. 2008

-7.164786e+02

-6.269188e+02

-5.373590e+02

-4.477991e+02

-3.582393e+02

-2.686795e+02

-1.791196e+02

-8.955983e+01

0.000000e+00

5.410934e+01

1.082187e+02

1.623280e+02

2.164374e+02

2.705467e+02

3.246560e+02

3.787654e+02

4.328747e+02

x

y

z

33

Jednostranne vazby (6)Napetı σy (klasicke vazby):

podlozka

uFEM 0.2.46

Set: 1: 1.000Result: s_y

23. 09. 2008

-1.534330e+03

-1.342539e+03

-1.150747e+03

-9.589562e+02

-7.671650e+02

-5.753737e+02

-3.835825e+02

-1.917912e+02

0.000000e+00

3.712790e+00

7.425580e+00

1.113837e+01

1.485116e+01

1.856395e+01

2.227674e+01

2.598953e+01

2.970232e+01

x

y

z

34

Jednostranne vazby (7)Napetı σ1 (klasicke vazby):

podlozka

uFEM 0.2.46


23. 09. 2008

-7.135765e+02

-6.243795e+02

-5.351824e+02

-4.459853e+02

-3.567883e+02

-2.675912e+02

-1.783941e+02

-8.919706e+01

0.000000e+00

5.488281e+01

1.097656e+02

1.646484e+02

2.195312e+02

2.744141e+02

3.292969e+02

3.841797e+02

4.390625e+02

x

y

z

35

Jednostranne vazby (8)Deformovany tvar (jednostranne vazby):

podlozka

uFEM 0.2.46

Set: 1: 1.000CS: CART

23. 09. 2008

-8.029127e+02

-7.025486e+02

-6.021845e+02

-5.018204e+02

-4.014564e+02

-3.010923e+02

-2.007282e+02

-1.003641e+02

0.000000e+00

4.536868e+01

9.073735e+01

1.361060e+02

1.814747e+02

2.268434e+02

2.722121e+02

3.175807e+02

3.629494e+02

x

y

z

36

Jednostranne vazby (9)Napetı σx (jednostranne vazby):

podlozka

uFEM 0.2.46

Set: 1: 1.000Result: s_x

23. 09. 2008

-8.084983e+02

-7.074360e+02

-6.063737e+02

-5.053114e+02

-4.042491e+02

-3.031869e+02

-2.021246e+02

-1.010623e+02

0.000000e+00

3.922044e+01

7.844088e+01

1.176613e+02

1.568818e+02

1.961022e+02

2.353226e+02

2.745431e+02

3.137635e+02

x

y

z

37

Jednostranne vazby (10)Napetı σy (jednostranne vazby):

podlozka

uFEM 0.2.46

Set: 1: 1.000Result: s_y

23. 09. 2008

-1.535040e+03

-1.343160e+03

-1.151280e+03

-9.594000e+02

-7.675200e+02

-5.756400e+02

-3.837600e+02

-1.918800e+02

0.000000e+00

6.736788e+00

1.347358e+01

2.021036e+01

2.694715e+01

3.368394e+01

4.042073e+01

4.715751e+01

5.389430e+01

x

y

z

38

Jednostranne vazby (11)Napetı σ1 (jednostranne vazby):

podlozka

uFEM 0.2.46


23. 09. 2008

-8.029127e+02

-7.025486e+02

-6.021845e+02

-5.018204e+02

-4.014564e+02

-3.010923e+02

-2.007282e+02

-1.003641e+02

0.000000e+00

4.536868e+01

9.073735e+01

1.361060e+02

1.814747e+02

2.268434e+02

2.722121e+02

3.175807e+02

3.629494e+02

x

y

z

39

Prıklad – ram (1)Rovnomerne zatızenı na obou polıch :

uFEM 0.2.78

Set: 1: 1.000

y

x

1.51113e+06

1.32224e+06

1.13335e+06

9.44455e+05

7.55564e+05

5.66673e+05

3.77782e+05

1.88891e+05

0.00000e+00

-2.51596e+05

-5.03192e+05

-7.54788e+05

-1.00638e+06

-1.25798e+06

-1.50958e+06

-1.76117e+06

-2.01277e+06

24. 09. 2017

Results

z

40

Prıklad – ram (2)Rovnomerne zatızenı v levem poli:

ram02

y

x

1.58274e+06

1.38490e+06

1.18706e+06

9.89215e+05

7.91372e+05

5.93529e+05

3.95686e+05

1.97843e+05

0.00000e+00

-8.36102e+04

-1.67220e+05

-2.50831e+05

-3.34441e+05

-4.18051e+05

-5.01661e+05

-5.85271e+05

-6.68881e+05

24. 09. 2017

ResultsSet: 1: 1.000

uFEM 0.2.78

z

41

Prıklad – ram (3)Rovnomerne zatızenı v levem poli a jednostranne vazby:

ram03

y

x

1.76342e+06

1.54299e+06

1.32256e+06

1.10214e+06

8.81708e+05

6.61281e+05

4.40854e+05

2.20427e+05

0.00000e+00

-4.38862e+04

-8.77724e+04

-1.31659e+05

-1.75545e+05

-2.19431e+05

-2.63317e+05

-3.07203e+05

-3.51090e+05

24. 09. 2017

ResultsSet: 1: 1.000

uFEM 0.2.78

z

42

Prıklad – ram (4)Porovnanı hlavnıch napetı σ1:

ram02

y

x

1.58274e+06

1.38490e+06

1.18706e+06

9.89215e+05

7.91372e+05

5.93529e+05

3.95686e+05

1.97843e+05

0.00000e+00

-8.36102e+04

-1.67220e+05

-2.50831e+05

-3.34441e+05

-4.18051e+05

-5.01661e+05

-5.85271e+05

-6.68881e+05

24. 09. 2017

Result: s_1Set: 1: 1.000

uFEM 0.2.78

z

ram03

y

x

1.76342e+06

1.54299e+06

1.32256e+06

1.10214e+06

8.81708e+05

6.61281e+05

4.40854e+05

2.20427e+05

0.00000e+00

-4.38862e+04

-8.77724e+04

-1.31659e+05

-1.75545e+05

-2.19431e+05

-2.63317e+05

-3.07203e+05

-3.51090e+05

24. 09. 2017

Result: s_1Set: 1: 1.000

uFEM 0.2.78

z

Pevne vazby:σ1,max ≈ 1.58MPa, jednostranne vazby:σ1,max ≈ 1.76MPa.

43

TE´MA 4: Technicka´ teorie skorˇepin (1)fast10.vsb.cz/brozovsky/data/nlm/p2.pdfza´kladnı´...

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