Linea elastica

1)

Eᵪ = 0 → X_A = 0

Eᵧ = 0 → Y_A + Y_B - qL = 0

            Y_A = qL/2   Y_B = qL/2

E_MA = 0 → qL (L/2) - Y_B(L) = 0 → Y_B(L) = qL²/2 → Y_B = qL²/2

2)

3)

H(x)_AB = qLx/2 - qx (x/2) = qLx/2 - qx²/2

4)

η^" = -q x²/2 + qL x/2

η^′ = ∫(q x²/2 - qL x/2) / EI dx

y' = ¹⁄_EI [^qx3⁄₆ - ^qLx2⁄₂] + c₁

y = ¹⁄_EI [^qx4⁄₂₄ - ^qLx3⁄₂] + c₁x + c₂

(5) NODO A x=0

η_A = 0

ψ_A = 0

φ_A = 0

NODO B x = L

η_B = 0

ψ_B ≠ 0

(6) { η_A = 0

{ η_B = 0 => ¹⁄_EI [^qL4⁄₂₄ - ^qL3⁄₁₂] + C₁L = 0 C₁L = -¹⁄_EI [ -^qL4⁄₂₄ ]

{ c₂ = 0

{ C₁ = ^qL3⁄_24EI

{ c₁ = ^qL3⁄_24EI 1/x

{ C₁ = ^qL3⁄_24EI

(7) ψ_A = ^qL3⁄_24EI

φ_B = ¹⁄_EI [^qL3⁄₆ - ^qL3⁄₄] + ^qL3⁄_24EI = ¹⁄_EI [^L-G⁄_24EI qL³] + ^ql3⁄_24EI - ^2qL3⁄_24EI + ^qL3⁄_24EI

= -^2t⁄_24EI qL³ - ^qL3⁄_24EI

Nodo A x=0

_A=0

_A=0

Nodo B x=l

_B⁰

_B⁰

_A=0

₁=0

_B=¹ /_6EI[ ³/ ₂ + ³]⁰

-⁴+3⁴)⁴/ ₃^8EI=⁴/ _3EI

(1)

ΣF_x = 0      X_A = 0

ΣF_y = 0      Y_A + ql = 0      Y_B = -ql

ΣM_A = 0      H_A + ql² - qlL = 0

          H_A = -ql² - ql²

          H_A = -2ql²

(2)

(3)

H(x)_AB = -2ql² + qlx

(4)

η'' = -qlx - 2ql²

η' = ∫ -qlx - 2ql² dx = ¹/_EI [^qlx2/₂ - 2ql^2x] + c₁

η = -¹/_EI [^qlx3/₆ - ^2ql2x2/₂] + c₁x + c₂

C₃=0

C₂= -9qL² / 6EI

C₄ = 9qL / 24EI

C₁ = 0

Calcolo M_A

M_A = 9qL / 24EI

Calcolo C_S

T_A = 0

M_B = -qL² / 2EI + 9qL² / 6EI = ( -3qL² + 9qL² ) / 6EI = -2qL² / 6

T_B = - (9q / EI + C₁) EI = - (9q / EI) EI = -9q

M_A = (qL² / 2EI) + ☐ + C₂ EI = - (0 + 0 - 9qL² / 6EI) EI = qL² / 6

qL² / 3

qL

A

B

A

B

M

qL₂

ψ^iv= C₁

T = ψ''' EI

M = ψ'' EI

ψ'' = C₁ x + C₂

M = C₁ x + C₂ x + C₃

ψ

ψ = C₁x³ / 6 - qC₂ x² + C₃ x + C₄

NOLO

x = 0

η_A = 0 T_A = 0

ψ_A ≠ 0 M_A = -qL²

NOLO B x = L

η_B = 0 T_B = 0

ψ_B = 0 T_B = 0

η = 0

ψ_A = 0

ψ_B = 0

C₆ = 0

qL³ / 6 + - (qC₂)(L² / 2) + C₉L = 0

C₂ = qL² / EI C₁ + C₃ = 0

C₃ = -C₁L² / 2 - qL³ / EI

C₁L³ / 6 + qL⁴ / 2 + (C₁L² − qC₃

EI)L = 0

C₁L³ / 6 +

qL³ / 2 C₁L³ / 2

- qL⁶

EI = 0

-lC₁L³ / 3 - -qC₆ / 2EI = 0

C₁L³ / 3 - qL⁴ / 2EI C₁ = qL⁴ / 2EI

C₆ = 0

C₁ = qL⁴ / 2EI

ψ_L = -3qL / 2EI - qL³

C = 3 - -(-3qL / 2EI) = 0 / 2 - qL³

EI = 0

calcolo ψ

ψ_A ≠ 0

ψ_A = -qL³ / 4EI

M_B = -[ (^qL⁄₂) + (^qL2⁄_2cEI) ] cEI = - (^qL2⁄_2c)

M_A = - ( ^qL2⁄_2cEI) cEI = - ^qL2⁄₂

In definitiva

^qL2⁄₂

S.S.

A

B

L

S.F. fittizio

XA = 0

XA = 0

EY = 0

YA + YB - qL = 0

YA = qL - YB

YA = qL/2

YB = qL/2

MA = qL²/2 - 4B L

YB = qL²/2

XA = 0

EX = 0

EA = -1/L

YA = 1/L

EY = 0

1 - 4B(L) = 0

YB = 1/L

Diagrammi e Momento

A

B

qL/2

x

T

M

A

B

HF(AB) = -x/L

MS(AB) = -qx²/2 + qLx/2

Lve = Lvi nelle iscr

Lve = F^SS x S^SF φB x i = -φB

LVI = ∫_A^B (1/EI) (MS(AB) - MF(AB)) dx

LVI = (1/EI) ∫₀^L (qx²/2 + qLx/2) (-x/2) dx

SS

A B

L E_x = 0 E_y = 0 E_H = 0

Diag nom

A B x

A B

M_S(AB) = m

LVe = LVi

dE_I

LVi = ∫₀^B M_S(AB)·H_F(AB) dx

LVi = 1/ε_I ∫₀^L (m)(x) dx

LVi = 1/ε_I ∫₀^L (m)(x) dx

2LVi = 1/a ^L [m x²/2]₀

LVi = 1/ε_I [m L²/2]

η/B = mL²/2ε_I

ESAME 16/08/2016

Determinare l'equazione della linea elastica, calcolare la reazione vincolare e tracciare i diagrammi delle azioni interne.

Ψ^IV = ^q/_EI poiché Ψ = q e il risultato Ψ^IV = -^q/_EI

Ψ^III = -^qX/_EI + C₁ T:- Ψ^III EI

Ψ^II = -^qX2/_2EI + C₁X + C₂ n= -Ψ^III EI

Ψ^I = -^qX3/_6EI + C₁X²/₂ + C₂X + C₃

Ψ = -^qX4/_24EI + C₁X³/₆ + C₂X²/₂ + C₃X + C₄

NODO A x=0

• Ψ_A = 0
• H_A = 0
• γ_A = 0
• T_A = 0

NODO B x=l

• Ψ_B = 0
• H_B ≠ 0
• γ_B ≠ 0
• T_B = -2q_L

Ψ_A = 0

γ_A = 0

Ψ_B = 0

-Ψ^III EI = -2qL

• C₃ = 0
• C₄ = 0

{ -^qL3/_6EI + C₁L²/₂ + C₂L = 0

{ -(-^qL/_EI + C₁) = -2qL/_EI

{ C₁ = 3q_L

{ -^qL3/_6EI + 3qL³/_2EI + C₂L = 0

C₂L - ^-qL/_EI qL 3

C₂ = -^qql3/_3EI

C₂ = -^qql3/_3EI

C₃ = 3ql