Esercizi Azioni Interne Strutture Piane Piu Aste

Schema n.001Meccanica dei Continui e delle Strutture

@ Adolfo Zavelani Rossi, Politecnico di Milano, versione 11.05

y,v,V,q

x,u,H,p

b

ϕ,W b 2b

A

B

C

D

E

q q

q

qAB = -q qBD = -q pDE = -q EJAB = EJ EJBC = EJ EJBD = EJ EJDE = EJ

Carichi e deformazioni date hanno verso efficace in disegno.Calcolare reazioni vincolari della struttura e delle aste.Tracciare i diagrammi delle azioni interne nelle aste.Esprimere la linea elastica delle aste.JYZ - xYZ - θYZ riferimento locale asta YZ con origine in Y.

Schema n.001

@ Adolfo Zavelani Rossi, Politecnico di Milano, versione 11.05 001

Schema n.001REAZIONI


q

11/6qb

5/6qb4/3qb2

A

Bqb

qb2

qb

B C

q

5/6qb

qb1/3qb2

7/6qb

qb

B

Dq

7/6qb

qb

7/6qb1/2qb2

D E

Schema n.001DEFORMATA E AZIONI INTERNE


2 qb4/EJ

A

B

C

D

E

1.5 qb

2 qb

1.5 qb2

Schema n.001PROCEDIMENTO E RISULTATI


DETERMINAZIONE DELLA DEFORMATA ELASTICACostanti di integrazione: ϕAB KAB ϕBC KBC ϕBD KBD ϕDE KDE

Relazioni di congruenzay’AB(b) - y’BD(0) = 0y’BC(0) - y’BD(0) = 0y’DE(b) = 0yAB(0) = 0yAB(b) - yBD(0) = 0yBC(b) = 0yBD(2b) = 0yDE(0) - yBC(0) = 0

MAB = 11/6Fx -1/2qx2

EJy" = 11/6Fx -1/2qx2

EJy’ = 11/12Fx2 -1/6qx3 +EJϕAB

EJy = 11/36Fx3 -1/24qx4 +EJϕABx +EJKAB

MBC = -Fx +FbEJy" = -Fx +FbEJy’ = -1/2Fx2 +Fbx +EJϕBC

EJy = -1/6Fx3 +1/2Fbx2 +EJϕBCx +EJKBC

MBD = 5/6Fx +1/3Fb -1/2qx2

EJy" = 5/6Fx +1/3Fb -1/2qx2

EJy’ = 5/12Fx2 +1/3Fbx -1/6qx3 +EJϕBD

EJy = 5/36Fx3 +1/6Fbx2 -1/24qx4 +EJϕBDx +EJKBD

MDE = Fx -1/2qx2

EJy" = Fx -1/2qx2

EJy’ = 1/2Fx2 -1/6qx3 +EJϕDE

EJy = 1/6Fx3 -1/24qx4 +EJϕDEx +EJKDE

Condizioni al contornoϕABb KAB ϕBCb KBC ϕBDb KBD ϕDEb KDE qb4/EJ

y’BA 1 0 0 0 -1 0 0 0 -3/4y’BC 0 0 1 0 -1 0 0 0 0y’ED 0 0 0 0 0 0 1 0 -1/3yAB 0 1 0 0 0 0 0 0 0yBA 1 1 0 0 0 -1 0 0 -19/72

=

yCB 0 0 1 1 0 0 0 0 -1/3yDB 0 0 0 0 2 1 0 0 -10/9yDE 0 0 0 -1 0 0 0 1 0

SoluzioneFb3/EJ

ϕABb -23/24ϕBCb -5/24ϕDEb -1/3KAB 0

ϕBDb -5/24=

KBD -25/36KBC -1/8KDE -1/8



REAZIONIVA = 11/6qb HC = qb VE = 7/6qb WE = 1/2qb2

HAB = 0VAB = 11/6qbWAB = 0HBA = 0VBA = -5/6qbWBA = 4/3qb2

HBC = -qbVBC = 0WBC = -qb2

HCB = qbVCB = 0WCB = 0

HBD = qbVBD = 5/6qbWBD = -1/3qb2

HDB = -qbVDB = 7/6qbWDB = 0

HDE = qbVDE = -7/6qbWDE = 0HED = 0VED = 7/6qbWED = 1/2qb2

DEFORMATA (coordinate locali)AB y(x)EJ = -23/24xqb3 +11/36x3qb -1/24x4qBA y(x)EJ = -25/36qb4 +5/24xqb3 +2/3x2qb2 -5/36x3qb -1/24x4qBC y(x)EJ = -1/8qb4 -5/24xqb3 +1/2x2qb2 -1/6x3qbCB y(x)EJ = -7/24xqb3 +1/6x3qbBD y(x)EJ = -25/36qb4 -5/24xqb3 +1/6x2qb2 +5/36x3qb -1/24x4qDB y(x)EJ = -19/24xqb3 +7/36x3qb -1/24x4qDE y(x)EJ = -1/8qb4 -1/3xqb3 +1/6x3qb -1/24x4qED y(x)EJ = -1/3qb4 +1/4x2qb2 -1/24x4q



y,v,V,q

x,u,H,p

b

ϕ,W b 2b

A

B

C

D

E

q q

q

qAB = -q qBD = -q pDE = -q EJAB = EJ EJBC = EJ EJBD = EJ EJDE = EJ


Schema n.001




q

qb

qb

qb1/2qb2

A

B9/4

B C

q

5/4qb

qb1/2qb2

3/4qb

qb

B

D

3/4qb

qb

3/4qb1/2qb2

D E



0.5 qb4/EJ

A

B

C

D

E

3 qb

1.5 qb

0.6 qb2




Relazioni di congruenzay’AB(b) - y’BD(0) = 0y’BC(0) - y’BD(0) = 0y’DE(b) = 0yAB(b) = 0yBC(0) = 0yBD(0) = 0yBD(2b) = 0yDE(0) = 0

MAB = -1/2qx2

EJy" = -1/2qx2

EJy’ = -1/6qx3 +EJϕAB

EJy = -1/24qx4 +EJϕABx +EJKAB

MBC = 0EJy" = 0EJy’ = EJϕBC

EJy = EJϕBCx +EJKBC

MBD = 5/4Fx -1/2Fb -1/2qx2

EJy" = 5/4Fx -1/2Fb -1/2qx2

EJy’ = 5/8Fx2 -1/2Fbx -1/6qx3 +EJϕBD

EJy = 5/24Fx3 -1/4Fbx2 -1/24qx4 +EJϕBDx +EJKBD

MDE = Fx -1/2qx2

EJy" = Fx -1/2qx2




y’BA 1 0 0 0 -1 0 0 0 1/6y’BC 0 0 1 0 -1 0 0 0 0y’ED 0 0 0 0 0 0 1 0 -1/3yBA 1 1 0 0 0 0 0 0 1/24yBC 0 0 0 1 0 0 0 0 0

=

yBD 0 0 0 0 0 1 0 0 0yDB 0 0 0 0 2 1 0 0 0yDE 0 0 0 0 0 0 0 1 0

SoluzioneFb3/EJ

ϕABb 1/6ϕBCb 0ϕDEb -1/3ϕBDb 0KBC 0

=

KBD 0KAB -1/8KDE 0



REAZIONIHA = qb VC = 9/4qb VE = 3/4qb WE = 1/2qb2

HAB = qbVAB = 0WAB = 0HBA = -qbVBA = qbWBA = -1/2qb2

HBC = 0VBC = -9/4qbWBC = 0HCB = 0VCB = 9/4qbWCB = 0

HBD = qbVBD = 5/4qbWBD = 1/2qb2



DEFORMATA (coordinate locali)AB y(x)EJ = -1/8qb4 +1/6xqb3 -1/24x4qBA y(x)EJ = -1/4x2qb2 +1/6x3qb -1/24x4qBC y(x)EJ = 0CB y(x)EJ = 0BD y(x)EJ = -1/4x2qb2 +5/24x3qb -1/24x4qDB y(x)EJ = -1/6xqb3 +1/8x3qb -1/24x4qDE y(x)EJ = -1/3xqb3 +1/6x3qb -1/24x4qED y(x)EJ = -5/24qb4 +1/4x2qb2 -1/24x4q



y,v,V,q

x,u,H,p

b

ϕ,W b 2b

A

B

C

D

E

q q

q

qAB = -q qBD = q pDE = -q EJAB = EJ EJBC = EJ EJBD = EJ EJDE = EJ


Schema n.001




q

1/2qb

1/2qb

A

Bqb

qb2

qb

B C

q

1/2qb

qbqb2

3/2qb

qb

B

D q

3/2qb

qb

3/2qb1/2qb2

D E



3 qb4/EJ

A

B

C

D

E

2 qb

2 qb

1.5 qb2





MAB = 1/2Fx -1/2qx2

EJy" = 1/2Fx -1/2qx2





MBD = -1/2Fx -Fb +1/2qx2

EJy" = -1/2Fx -Fb +1/2qx2

EJy’ = -1/4Fx2 -Fbx +1/6qx3 +EJϕBD

EJy = -1/12Fx3 -1/2Fbx2 +1/24qx4 +EJϕBDx +EJKBD

MDE = Fx -1/2qx2

EJy" = Fx -1/2qx2





=

yCB 0 0 1 1 0 0 0 0 -1/3yDB 0 0 0 0 2 1 0 0 2yDE 0 0 0 -1 0 0 0 1 0

SoluzioneFb3/EJ

ϕABb 43/72ϕBCb 49/72ϕDEb -1/3KAB 0

ϕBDb 49/72=

KBD 23/36KBC -73/72KDE -73/72



REAZIONIVA = 1/2qb HC = qb VE = -3/2qb WE = 1/2qb2

HAB = 0VAB = 1/2qbWAB = 0HBA = 0VBA = 1/2qbWBA = 0



HBD = qbVBD = -1/2qbWBD = qb2

HDB = -qbVDB = -3/2qbWDB = 0

HDE = qbVDE = 3/2qbWDE = 0HED = 0VED = -3/2qbWED = 1/2qb2

DEFORMATA (coordinate locali)AB y(x)EJ = 43/72xqb3 +1/12x3qb -1/24x4qBA y(x)EJ = 23/36qb4 -49/72xqb3 +1/12x3qb -1/24x4qBC y(x)EJ = -73/72qb4 +49/72xqb3 +1/2x2qb2 -1/6x3qbCB y(x)EJ = -85/72xqb3 +1/6x3qbBD y(x)EJ = 23/36qb4 +49/72xqb3 -1/2x2qb2 -1/12x3qb +1/24x4qDB y(x)EJ = 71/72xqb3 -1/4x3qb +1/24x4qDE y(x)EJ = -73/72qb4 -1/3xqb3 +1/6x3qb -1/24x4qED y(x)EJ = -11/9qb4 +1/4x2qb2 -1/24x4q



y,v,V,q

x,u,H,p

b

ϕ,W b 2b

A

B

C

D

E

q q

q

qAB = -q qBD = q pDE = -q EJAB = EJ EJBC = EJ EJBD = EJ EJDE = EJ


Schema n.001




q

qb

qb

qb1/2qb2

A

B1/4

B C

q

3/4qb

qb1/2qb2

5/4qb

qb

B

D

5/4qb

qb

5/4qb1/2qb2

D E



2 qb4/EJ

A

B

C

D

E

1.5 qb

1.5 qb

1 qb2





MAB = -1/2qx2

EJy" = -1/2qx2





MBD = -3/4Fx -1/2Fb +1/2qx2

EJy" = -3/4Fx -1/2Fb +1/2qx2

EJy’ = -3/8Fx2 -1/2Fbx +1/6qx3 +EJϕBD

EJy = -1/8Fx3 -1/4Fbx2 +1/24qx4 +EJϕBDx +EJKBD

MDE = Fx -1/2qx2

EJy" = Fx -1/2qx2




y’BA 1 0 0 0 -1 0 0 0 1/6y’BC 0 0 1 0 -1 0 0 0 0y’ED 0 0 0 0 0 0 1 0 -1/3yBA 1 1 0 0 0 0 0 0 1/24yBC 0 0 0 1 0 0 0 0 0

=

yBD 0 0 0 0 0 1 0 0 0yDB 0 0 0 0 2 1 0 0 4/3yDE 0 0 0 0 0 0 0 1 0

SoluzioneFb3/EJ

ϕABb 5/6ϕBCb 2/3ϕDEb -1/3ϕBDb 2/3KBC 0

=

KBD 0KAB -19/24KDE 0



REAZIONIHA = qb VC = 1/4qb VE = -5/4qb WE = 1/2qb2

HAB = qbVAB = 0WAB = 0HBA = -qbVBA = qbWBA = -1/2qb2


HBD = qbVBD = -3/4qbWBD = 1/2qb2

HDB = -qbVDB = -5/4qbWDB = 0

HDE = qbVDE = 5/4qbWDE = 0HED = 0VED = -5/4qbWED = 1/2qb2

DEFORMATA (coordinate locali)AB y(x)EJ = -19/24qb4 +5/6xqb3 -1/24x4qBA y(x)EJ = -2/3xqb3 -1/4x2qb2 +1/6x3qb -1/24x4qBC y(x)EJ = 2/3xqb3

CB y(x)EJ = 2/3qb4 -2/3xqb3

BD y(x)EJ = 2/3xqb3 -1/4x2qb2 -1/8x3qb +1/24x4qDB y(x)EJ = 1/2xqb3 -5/24x3qb +1/24x4qDE y(x)EJ = -1/3xqb3 +1/6x3qb -1/24x4qED y(x)EJ = -5/24qb4 +1/4x2qb2 -1/24x4q



y,v,V,q

x,u,H,p

b

ϕ,W b 2b

A

B

C

D

E

q q

q

qAB = -q qBD = -q pDE = q EJAB = EJ EJBC = EJ EJBD = EJ EJDE = EJ


Schema n.001




q

7/6qb

1/6qb2/3qb2

A

Bqb

qb2

qb

B C

q

1/6qb

qb5/3qb2

11/6qb

qb

B

Dq

11/6qb

qb

11/6qb1/2qb2

D E



4 qb4/EJ

A

B

C

D

E

2 qb

2 qb

2 qb2





MAB = 7/6Fx -1/2qx2

EJy" = 7/6Fx -1/2qx2



MBC = Fx -FbEJy" = Fx -FbEJy’ = 1/2Fx2 -Fbx +EJϕBC

EJy = 1/6Fx3 -1/2Fbx2 +EJϕBCx +EJKBC

MBD = 1/6Fx +5/3Fb -1/2qx2

EJy" = 1/6Fx +5/3Fb -1/2qx2



MDE = -Fx +1/2qx2

EJy" = -Fx +1/2qx2

EJy’ = -1/2Fx2 +1/6qx3 +EJϕDE

EJy = -1/6Fx3 +1/24qx4 +EJϕDEx +EJKDE


y’BA 1 0 0 0 -1 0 0 0 -5/12y’BC 0 0 1 0 -1 0 0 0 0y’ED 0 0 0 0 0 0 1 0 1/3yAB 0 1 0 0 0 0 0 0 0yBA 1 1 0 0 0 -1 0 0 -11/72

=

yCB 0 0 1 1 0 0 0 0 1/3yDB 0 0 0 0 2 1 0 0 -26/9yDE 0 0 0 -1 0 0 0 1 0

SoluzioneFb3/EJ

ϕABb -31/24ϕBCb -7/8ϕDEb 1/3KAB 0

ϕBDb -7/8=

KBD -41/36KBC 29/24KDE 29/24



REAZIONIVA = 7/6qb HC = -qb VE = 11/6qb WE = -1/2qb2


HBC = qbVBC = 0WBC = qb2

HCB = -qbVCB = 0WCB = 0

HBD = -qbVBD = 1/6qbWBD = -5/3qb2

HDB = qbVDB = 11/6qbWDB = 0

HDE = -qbVDE = -11/6qbWDE = 0HED = 0VED = 11/6qbWED = -1/2qb2

DEFORMATA (coordinate locali)AB y(x)EJ = -31/24xqb3 +7/36x3qb -1/24x4qBA y(x)EJ = -41/36qb4 +7/8xqb3 +1/3x2qb2 -1/36x3qb -1/24x4qBC y(x)EJ = 29/24qb4 -7/8xqb3 -1/2x2qb2 +1/6x3qbCB y(x)EJ = 11/8xqb3 -1/6x3qbBD y(x)EJ = -41/36qb4 -7/8xqb3 +5/6x2qb2 +1/36x3qb -1/24x4qDB y(x)EJ = -35/24xqb3 +11/36x3qb -1/24x4qDE y(x)EJ = 29/24qb4 +1/3xqb3 -1/6x3qb +1/24x4qED y(x)EJ = 17/12qb4 -1/4x2qb2 +1/24x4q



y,v,V,q

x,u,H,p

b

ϕ,W b 2b

A

B

C

D

E

q q

q

qAB = -q qBD = -q pDE = q EJAB = EJ EJBC = EJ EJBD = EJ EJDE = EJ


Schema n.001




q

qb

qb

qb1/2qb2

A

B9/4

B C

q

5/4qb

qb1/2qb2

3/4qb

qb

B

D

3/4qb

qb

3/4qb1/2qb2

D E



0.5 qb4/EJ

A

B

C

D

E

3 qb

1.5 qb

0.6 qb2





MAB = -1/2qx2

EJy" = -1/2qx2





MBD = 5/4Fx -1/2Fb -1/2qx2

EJy" = 5/4Fx -1/2Fb -1/2qx2



MDE = -Fx +1/2qx2

EJy" = -Fx +1/2qx2




y’BA 1 0 0 0 -1 0 0 0 1/6y’BC 0 0 1 0 -1 0 0 0 0y’ED 0 0 0 0 0 0 1 0 1/3yBA 1 1 0 0 0 0 0 0 1/24yBC 0 0 0 1 0 0 0 0 0

=

yBD 0 0 0 0 0 1 0 0 0yDB 0 0 0 0 2 1 0 0 0yDE 0 0 0 0 0 0 0 1 0

SoluzioneFb3/EJ

ϕABb 1/6ϕBCb 0ϕDEb 1/3ϕBDb 0KBC 0

=

KBD 0KAB -1/8KDE 0



REAZIONIHA = -qb VC = 9/4qb VE = 3/4qb WE = -1/2qb2

HAB = -qbVAB = 0WAB = 0HBA = qbVBA = qbWBA = -1/2qb2


HBD = -qbVBD = 5/4qbWBD = 1/2qb2



DEFORMATA (coordinate locali)AB y(x)EJ = -1/8qb4 +1/6xqb3 -1/24x4qBA y(x)EJ = -1/4x2qb2 +1/6x3qb -1/24x4qBC y(x)EJ = 0CB y(x)EJ = 0BD y(x)EJ = -1/4x2qb2 +5/24x3qb -1/24x4qDB y(x)EJ = -1/6xqb3 +1/8x3qb -1/24x4qDE y(x)EJ = 1/3xqb3 -1/6x3qb +1/24x4qED y(x)EJ = 5/24qb4 -1/4x2qb2 +1/24x4q



y,v,V,q

x,u,H,p

b

ϕ,W b 2b

A

B

C

D

E

q q

q

qAB = q qBD = -q pDE = -q EJAB = EJ EJBC = EJ EJBD = EJ EJDE = EJ


Schema n.001




q

1/6qb

7/6qb2/3qb2

A

Bqb

qb2

qb

B C

q

7/6qb

qb1/3qb2

5/6qb

qb

B

D q

5/6qb

qb

5/6qb1/2qb2

D E



1.2 qb4/EJ

A

B

C

D

E

1.2 qb

1.5 qb

1.2 qb2





MAB = 1/6Fx +1/2qx2

EJy" = 1/6Fx +1/2qx2

EJy’ = 1/12Fx2 +1/6qx3 +EJϕAB

EJy = 1/36Fx3 +1/24qx4 +EJϕABx +EJKAB



MBD = 7/6Fx -1/3Fb -1/2qx2

EJy" = 7/6Fx -1/3Fb -1/2qx2



MDE = Fx -1/2qx2

EJy" = Fx -1/2qx2





=

yCB 0 0 1 1 0 0 0 0 -1/3yDB 0 0 0 0 2 1 0 0 -2/9yDE 0 0 0 -1 0 0 0 1 0

SoluzioneFb3/EJ

ϕABb -19/72ϕBCb -1/72ϕDEb -1/3KAB 0

ϕBDb -1/72=

KBD -7/36KBC -23/72KDE -23/72



REAZIONIVA = 1/6qb HC = qb VE = 5/6qb WE = 1/2qb2




HBD = qbVBD = 7/6qbWBD = 1/3qb2



DEFORMATA (coordinate locali)AB y(x)EJ = -19/72xqb3 +1/36x3qb +1/24x4qBA y(x)EJ = -7/36qb4 +1/72xqb3 +1/3x2qb2 -7/36x3qb +1/24x4qBC y(x)EJ = -23/72qb4 -1/72xqb3 +1/2x2qb2 -1/6x3qbCB y(x)EJ = -35/72xqb3 +1/6x3qbBD y(x)EJ = -7/36qb4 -1/72xqb3 -1/6x2qb2 +7/36x3qb -1/24x4qDB y(x)EJ = -23/72xqb3 +5/36x3qb -1/24x4qDE y(x)EJ = -23/72qb4 -1/3xqb3 +1/6x3qb -1/24x4qED y(x)EJ = -19/36qb4 +1/4x2qb2 -1/24x4q



y,v,V,q

x,u,H,p

b

ϕ,W b 2b

A

B

C

D

E

q q

q

qAB = q qBD = -q pDE = -q EJAB = EJ EJBC = EJ EJBD = EJ EJDE = EJ


Schema n.001




q

qb

qb

qb1/2qb2

A

B1/4

B C

q

3/4qb

qb1/2qb2

5/4qb

qb

B

D

5/4qb

qb

5/4qb1/2qb2

D E



2 qb4/EJ

A

B

C

D

E

1.5 qb

1.5 qb

1 qb2





MAB = 1/2qx2

EJy" = 1/2qx2

EJy’ = 1/6qx3 +EJϕAB

EJy = 1/24qx4 +EJϕABx +EJKAB



MBD = 3/4Fx +1/2Fb -1/2qx2

EJy" = 3/4Fx +1/2Fb -1/2qx2



MDE = Fx -1/2qx2

EJy" = Fx -1/2qx2




y’BA 1 0 0 0 -1 0 0 0 -1/6y’BC 0 0 1 0 -1 0 0 0 0y’ED 0 0 0 0 0 0 1 0 -1/3yBA 1 1 0 0 0 0 0 0 -1/24yBC 0 0 0 1 0 0 0 0 0

=

yBD 0 0 0 0 0 1 0 0 0yDB 0 0 0 0 2 1 0 0 -4/3yDE 0 0 0 0 0 0 0 1 0

SoluzioneFb3/EJ

ϕABb -5/6ϕBCb -2/3ϕDEb -1/3ϕBDb -2/3KBC 0

=

KBD 0KAB 19/24KDE 0



REAZIONIHA = qb VC = -1/4qb VE = 5/4qb WE = 1/2qb2

HAB = qbVAB = 0WAB = 0HBA = -qbVBA = -qbWBA = 1/2qb2

HBC = 0VBC = 1/4qbWBC = 0HCB = 0VCB = -1/4qbWCB = 0

HBD = qbVBD = 3/4qbWBD = -1/2qb2



DEFORMATA (coordinate locali)AB y(x)EJ = 19/24qb4 -5/6xqb3 +1/24x4qBA y(x)EJ = 2/3xqb3 +1/4x2qb2 -1/6x3qb +1/24x4qBC y(x)EJ = -2/3xqb3

CB y(x)EJ = -2/3qb4 +2/3xqb3

BD y(x)EJ = -2/3xqb3 +1/4x2qb2 +1/8x3qb -1/24x4qDB y(x)EJ = -1/2xqb3 +5/24x3qb -1/24x4qDE y(x)EJ = -1/3xqb3 +1/6x3qb -1/24x4qED y(x)EJ = -5/24qb4 +1/4x2qb2 -1/24x4q



y,v,V,q

x,u,H,p

b

ϕ,W b 2b

A

B

C

D

E

q q

q

qAB = q qBD = -q pDE = q EJAB = EJ EJBC = EJ EJBD = EJ EJDE = EJ


Schema n.001




q

1/2qb

1/2qb

A

Bqb

qb2

qb

B C

q

1/2qb

qbqb2

3/2qb

qb

B

D q

3/2qb

qb

3/2qb1/2qb2

D E



3 qb4/EJ

A

B

C

D

E

2 qb

2 qb

1.5 qb2





MAB = -1/2Fx +1/2qx2

EJy" = -1/2Fx +1/2qx2

EJy’ = -1/4Fx2 +1/6qx3 +EJϕAB

EJy = -1/12Fx3 +1/24qx4 +EJϕABx +EJKAB

MBC = Fx -FbEJy" = Fx -FbEJy’ = 1/2Fx2 -Fbx +EJϕBC

EJy = 1/6Fx3 -1/2Fbx2 +EJϕBCx +EJKBC

MBD = 1/2Fx +Fb -1/2qx2

EJy" = 1/2Fx +Fb -1/2qx2

EJy’ = 1/4Fx2 +Fbx -1/6qx3 +EJϕBD


MDE = -Fx +1/2qx2

EJy" = -Fx +1/2qx2




y’BA 1 0 0 0 -1 0 0 0 1/12y’BC 0 0 1 0 -1 0 0 0 0y’ED 0 0 0 0 0 0 1 0 1/3yAB 0 1 0 0 0 0 0 0 0yBA 1 1 0 0 0 -1 0 0 1/24

=

yCB 0 0 1 1 0 0 0 0 1/3yDB 0 0 0 0 2 1 0 0 -2yDE 0 0 0 -1 0 0 0 1 0

SoluzioneFb3/EJ

ϕABb -43/72ϕBCb -49/72ϕDEb 1/3KAB 0

ϕBDb -49/72=

KBD -23/36KBC 73/72KDE 73/72



REAZIONIVA = -1/2qb HC = -qb VE = 3/2qb WE = -1/2qb2

HAB = 0VAB = -1/2qbWAB = 0HBA = 0VBA = -1/2qbWBA = 0

HBC = qbVBC = 0WBC = qb2

HCB = -qbVCB = 0WCB = 0

HBD = -qbVBD = 1/2qbWBD = -qb2



DEFORMATA (coordinate locali)AB y(x)EJ = -43/72xqb3 -1/12x3qb +1/24x4qBA y(x)EJ = -23/36qb4 +49/72xqb3 -1/12x3qb +1/24x4qBC y(x)EJ = 73/72qb4 -49/72xqb3 -1/2x2qb2 +1/6x3qbCB y(x)EJ = 85/72xqb3 -1/6x3qbBD y(x)EJ = -23/36qb4 -49/72xqb3 +1/2x2qb2 +1/12x3qb -1/24x4qDB y(x)EJ = -71/72xqb3 +1/4x3qb -1/24x4qDE y(x)EJ = 73/72qb4 +1/3xqb3 -1/6x3qb +1/24x4qED y(x)EJ = 11/9qb4 -1/4x2qb2 +1/24x4q



y,v,V,q

x,u,H,p

b

ϕ,W b 2b

A

B

C

D

E

q q

q

qAB = q qBD = -q pDE = q EJAB = EJ EJBC = EJ EJBD = EJ EJDE = EJ


Schema n.001




q

qb

qb

qb1/2qb2

A

B1/4

B C

q

3/4qb

qb1/2qb2

5/4qb

qb

B

D

5/4qb

qb

5/4qb1/2qb2

D E



2 qb4/EJ

A

B

C

D

E

1.5 qb

1.5 qb

1 qb2





MAB = 1/2qx2

EJy" = 1/2qx2

EJy’ = 1/6qx3 +EJϕAB

EJy = 1/24qx4 +EJϕABx +EJKAB



MBD = 3/4Fx +1/2Fb -1/2qx2

EJy" = 3/4Fx +1/2Fb -1/2qx2



MDE = -Fx +1/2qx2

EJy" = -Fx +1/2qx2




y’BA 1 0 0 0 -1 0 0 0 -1/6y’BC 0 0 1 0 -1 0 0 0 0y’ED 0 0 0 0 0 0 1 0 1/3yBA 1 1 0 0 0 0 0 0 -1/24yBC 0 0 0 1 0 0 0 0 0

=

yBD 0 0 0 0 0 1 0 0 0yDB 0 0 0 0 2 1 0 0 -4/3yDE 0 0 0 0 0 0 0 1 0

SoluzioneFb3/EJ

ϕABb -5/6ϕBCb -2/3ϕDEb 1/3ϕBDb -2/3KBC 0

=

KBD 0KAB 19/24KDE 0



REAZIONIHA = -qb VC = -1/4qb VE = 5/4qb WE = -1/2qb2

HAB = -qbVAB = 0WAB = 0HBA = qbVBA = -qbWBA = 1/2qb2

HBC = 0VBC = 1/4qbWBC = 0HCB = 0VCB = -1/4qbWCB = 0

HBD = -qbVBD = 3/4qbWBD = -1/2qb2



DEFORMATA (coordinate locali)AB y(x)EJ = 19/24qb4 -5/6xqb3 +1/24x4qBA y(x)EJ = 2/3xqb3 +1/4x2qb2 -1/6x3qb +1/24x4qBC y(x)EJ = -2/3xqb3

CB y(x)EJ = -2/3qb4 +2/3xqb3

BD y(x)EJ = -2/3xqb3 +1/4x2qb2 +1/8x3qb -1/24x4qDB y(x)EJ = -1/2xqb3 +5/24x3qb -1/24x4qDE y(x)EJ = 1/3xqb3 -1/6x3qb +1/24x4qED y(x)EJ = 5/24qb4 -1/4x2qb2 +1/24x4q

Esercizi Azioni Interne Strutture Piane Piu Aste

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