Statica si Stabilitatea Structurilor III · Statica si Stabilitatea Structurilor III Note de curs: ...

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Statica si Stabilitatea Structurilor III

Note de curs: www.cosminchiorean.com

Calculul Geometric Neliniar al Structurilor

Ecuatiile de echilibru a barei drepte in calculul geometric neliniar

0

)()()( dx

x

xNxNxNFx

tconsxNx

xNtan)(0

)(

0

)()()( qdxdx

x

xVxVxVFy

0)(

)(

2)(

)()(

2

xMdxx

xMdxqxMdx

x

xvNdxxVM t

qx

xV

)(

x

xvNxV

x

xM t

)()(

)(

xvxvxvt )()( 0

x

xv

x

xvNxV

x

xM )()()(

)( 0

Ecuatiile de echilibru a barei drepte in calculul geometric neliniar

Ecuatiile de echilibru a barei drepte in calculul geometric neliniar

x

xv

x

xvNxV

x

xMxT

)()()(

)()( 0

2

2 )()(

x

xvEIxM

2

0

2

2

2 )()(

)(

x

xvNqxM

EI

N

x

xM

MM

NVV

VNN

sincos

sincos

MM

NVV

VNN

1cos sin

x

vNV

x

MTV t

Ecuatiile de echilibru a barei drepte in calculul geometric neliniar

Ecuatiile de echilibru ale barei drepte in calculul geometric neliniar: Efectul deformatiilor de lunecare

)()()()( 0 xvxvxvxv SHBt

x

xv

x

xv

x

xv

x

xv SHBt

)()()( 0

x

vNV

x

MT tE

x

v

x

vNVT B

H0

Ecuatiile de echilibru a barei drepte in calculul geometric neliniar: Efectul deformatiilor de lunecare

Ecuatiile de echilibru a barei drepte in calculul geometric neliniar: Efectul deformatiilor de lunecare-Modelul Engesser

x

v

x

v

x

vN

x

M

x

vN

x

MVM SHBt 00

qx

v

x

v

x

vN

x

M

x

vN

x

Mq

x

VF SHBty

2

0

2

2

2

2

2

2

2

2

2

2

2

0

2

2 )()(

x

xvEIxM B

x

xvGAT SH

sE

)(

2

2

2

2 )(

x

xM

x

xvGA SH

s

2

0

2

2

2

1x

vNqxM

EI

N

x

xM

GA

N

s

ss GA

Nx

vNqxM

GA

NEI

N

x

xM

1

1

1

12

0

2

2

2

x

xMTE

2

2

x

xM

x

TE

Ecuatiile de echilibru a barei drepte in calculul geometric neliniar: Efectul deformatiilor de lunecare-Modelul Haringx

x

v

x

vNVT B

H0

x

xvGAT SH

sH

)(x

xvGA

x

v

x

vNV SH

sB

)(0

2

2

2

0

2

2

2 )(

x

xvGA

x

v

x

vN

x

V SH

sB

x

v

x

v

x

v

x

v

EI

M

x

v

qx

V

BtSH

B

0

2

2

2

2

2

2 )()(

x

xvNq

x

xM t

2

0

22

2

2

0

2 )(

)(

)(x

v

EI

xM

N

x

xMq

GAx

vNxM

EI

Nq s

ss GA

N

x

vNqxM

GA

N

EI

N

x

xM1)(1

)(2

0

2

2

2

Ecuatiile de echilibru ale barei drepte in calculul geometric neliniar: Efectul deformatiilor de lunecare

Integrarea ecuatiilor diferentiale de echilibru pentru bara dreapta Formularea in momente incovoietoare

2

0

2

2

2 )()(

)(

dx

xvdNqxM

EI

N

dx

xMd

2

0

2

2

)(

dx

xvdNqx

EI

N

)(int0,)()(

)(0,)()(

2

2

2

2

2

2

indereNxxMdx

xMd

ecompresiunNxxMdx

xMd

)()( xMxMxM po 0)()( 2

2

2

xMdx

xMd

022 r

)(int0,

)(0,2,1

indereN

ecompresiunNir

xCxCxM o sincos 21 xCxCeCeCxM xx

o sinhcosh 2121

200

)(4

L

xLxvxv m

tcons

L

Nvq

dx

xvdNqx m tan

8)(2

0

2

0

2

0,

0,

)(

2

2

N

N

xM p

0,sinhcosh)(

0,sincos)(

221

221

NxCxCxM

NxCxCxM

Compresiune Intindere

x

L

Lx

L

xMx

L

LxMxM ji

sin

sin

1coscos1

sin

sinsin

sin

coscos)(

2

j

i

MLCLCLM

MCM

221

21

sincos)(

)0(

2cos

2cos

1sin

sin

sin

1sin

)(2 L

xL

L

xM

L

L

xL

MxM ji

EI

NLL

2cos

2cos

1sin

sin

sin

1sin

)(2

2

L

x

LL

x

ML

x

MxM ji

200

)(4

L

xLxvxv m

2cos

2cos

1sin

sin

sin

1sin

)(2

2

L

x

LL

x

ML

x

MxM ji

Integrarea ecuatiilor diferentiale de echilibru pentru bara dreapta Formularea in deplasari

2

0

2

2

2

2

2

2

2 )()()()(

dx

xvd

dx

xvdNq

dx

xvdNq

dx

xMd t

2

2 )()(

dx

xvdEIxM

2

0

2

2

2

4

4 )(1)()(

dx

xvdNq

EIdx

xvd

EI

N

dx

xvd

2

0

2

2

)(1

dx

xvdNq

EIx

EI

N

)(int0,)()(

)(0,)()(

2

22

4

4

2

22

4

4

indereNxdx

xvd

dx

xvd

ecompresiunNxdx

xvd

dx

xvd

EI

q

L

Nvq

EIdx

xvd

EI

N

dx

xvd m '81)()(2

0

2

2

4

4

EI

q

xwdx

xvd

'

)()(

2

2

)()( 2

2

2

xwdx

xwd

0,sinh'cosh')(

0,sin'cos')(

221

221

NxCxCxw

NxCxCxw

3221 cos1'sin

1'3)(

)(CxxCxCCdxxw

dx

xdv

43

2

22221

3221

2sin

1'cos

1'

4cos1'sin

1')(

CxCx

xCxC

CdxCxxCxCxv

2sincos)(

2

24321

xCxCxCxCxv

2321 cossin)(

xCxCxC

dx

xdv

22

2

1

2

2

2

sincos)()(

xCxC

dx

xvd

EI

xM

xCxCdx

xvd

EI

xT cossin

)()(2

3

1

3

3

3

dx

xdv

dx

xdvNxV

dx

xdMxT

)()()(

)()( 0

dx

xdvN

dx

xdvNxTxV

)()()()( 0

dx

xdv

EI

N

dx

xdv

EI

N

EI

xT

EI

xV )()()()( 0

xLL

vxC

EI

xV m 24

3)(

2

022

TxVxMxxvx )()()()( z

xLL

vEIEIx

x

x

C

C

C

C

EI

xEIxEI

xx

xxx

xV

xM

x

xv

m 24

2

000

00sincos

01cossin

1sincos

)(

)(

)(

)(

2

02

2

2

2

2

4

3

2

1

2

22

xxx qCBz )()(

0)0()0(1 qzBC

0)0(0)0()0(1 qzUqzBBz xxx

Bara dreapta cu discontinuitati in lungul ei

LxlxhxgCxfCxM

lxxhxgCxfCxM

1224232

1112111

),()()()(

0),()()()(

)(')(')(')(')(')('

)()()()()()(

)()()(

)0()0()0(

1212412311112111

1212412311112111

22423

11211

lhlgClfClhlgClfC

lhlgClfClhlgClfC

MLhLgCLfC

MhgCfC

j

i

)(')('

)()(

)(

)0(

0

0

)(')(')(')('

)()()()(

)()(00

00)0()0(

1112

1112

2

1

4

3

2

1

12121111

12121111

22

11

lhlh

lhlh

Lh

h

M

M

C

C

C

C

lglflglf

lglflglf

LgLf

gf

j

i

hmfc

hmfc 1

4

3

2

1

444

1

343

1

242

1

141

1

42

1

41

1

4

434

1

333

1

232

1

131

1

32

1

31

1

3

424

1

323

1

222

1

121

1

22

1

21

1

2

414

1

313

1

212

1

111

1

12

1

11

1

1

H

ji

H

ji

H

ji

H

ji

hhhhMMC

hhhhMMC

hhhhMMC

hhhhMMC

ffffff

ffffff

ffffff

ffffff

LxlxhxgHxfHxgxfMxgxfMxM

lxxhxgHxfHxgxfMxgxfMxM

xCxB

j

xB

i

xCxB

j

xB

i

1

)(

22423

)(

242

1

232

1

)(

241

1

231

1

2

1

)(

11211

)(

122

1

112

1

)(

121

1

111

1

1

,)()()()()()()()(

0,)()()()()()()()(

22,22,1

11,21,1

ffff

ffff

)()()()( 21 xCxBMxBMxM ji

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