Esercizi d'esame svolti di Scienza delle Costruzioni 2

TESTO ESAME APRILE 2020

m = 12

n = 3.4 = 12

m = n = 6

sistema isostatico

Vincoli Esterni

• Σx = 0 → yC - 2PL + x3 + xA = 0
• ΣMA = 0 → -yB - yB L + PL² + 3/2 P2 + 3PL²/3 + 2PL² - 4yCL = 0
• Σy = 0 → yC - PL + yB + yA = 0

Vincoli Interni

1. • Σx = 0 → yC - 2PL = 0 → xC = 2PL
   • Σy = 0 → 4 - yC - y0D1 = 0 → xC = 3PL/2
   • ΣMC = 0 → 2PL² - M0 - y0D1L = 0
2. • Σx = 0 → yC - xE = 0 → xE = 0
   • Σy = 0 → 7 - y0D2 = 0 → y0D2 = yE = PL²/2
   • ΣM0 = 0 → M0 - yEL = 0 → M0 = -yEL = -PL²/2

3)

• Σx=0 → x_G-x_F=0 → -γx_B=0
• Σy=0 → y_E - P_L + y_B - y_F=0 → y_F = ^{P · L}/₂
• ΣMF=0 → ^3PL/₂ y_G L + P^L2 - y_B L + x ^3L/₂ =0 → y_B= ^3PL/₂

4)

• Σx=0 → x_A+x_F=0 → x_A=0
• Σy=0 → y_A+y_F=0 → y_A=-PL
• ΣMA=0 → x_FL=0 → x_F=0

5)

• ΣH_C=0 → 2rL²-PL² + 2ψ_EL=0 → ψ_E= + ^PL/₂₄ + ^PL/₂

ESPLE

DIAGRAMMI

Ⓒ

T

VINCOLI INTERNI Pi

1.

• Σx = 0 → 1 + x_D1 = 0
• Σy = 0 → -y_A + y_B1 = 0 → y_B1 = y_A
• ΣM_D1 = 0 → 1 - y_AL = 0 → y_A = ¹/_L

2.

• Σx = 0 → x_D2 + x_C = 0 → x_C = 0
• Σy = 0 → -y_B2 + y_C = 0 → y_B2 = 0
• ΣM_D2 = 0 → y_C = 0 → y_C = 0

3.

• Σx = 0 → x_D2 + x_B = 0 → x_B = 0
• Σy = 0 → y_D2 + y_B = 0 → y_B = PL
• ΣM_D2 = 0 → x_BL = 0 → x_B = 0

x_D1 + x_D2 + x_B = 0 → x_B = -x_D1 - x_D2 = 0 → 0 = 0

y_D1 + y_D2 + y_B = 0 → -PL + y_D2 + 0 = 0 → y_D2 = PL

ESPL

• 1. T = 0
• M_a = PL²

• 2. T = PL
• N = PL x + c → N(0) = PL² x = c x e²
• M = PL x PL²

1.2)

θ₂ = 2L = θ₃ - 2L → θ₂ = θ₃

θ₁ = S / 2L = θ₂

θ_D = θ₃ · L = S / 2L = S / 2

θ_E = θ₁ · L = S / 2

θ_A = θ₂ · L = S / 2

y_Σ = -2PL · S / X - 2PL · S / X - PL · S / 2 + PL · S² / 24

y = ( -2PL - PL / 2 + PL / 2 ) = +2PL

∑x = 0 → Xe = 0

∑y = 0 → Ye + Yd = -ye = 2PL/3

∑He = 0 → L + Hd = 0 → Hd = + 2/3 PL²

∑x = 0 → Xf = 0

∑y = 0 → -Yf - 2PL + Yf = 0 → Yd = -2/3 PL

∑Hd = 0 → 2yfL - 2d = 0 - Yd = 2/3 PL

∑x = 0 → Xc - Xa/ρL = 0 → Xc = 0

∑Mc = 0 → PL² + 2PL² + 2YfaL + 2Xd/2L² + 2Ya0L + Hd = 0

∑Hee = 0 → 4PL² + 3Yfa/1 = 0 → Yf = 4 PL/3

ESPLOSO

1. T = P(Xc - Xb - St(0) = 2PL/3

M = -Px/2 + Xc - Xb - St(0) = 2PL² /3

H(l_c) = PL² + 2PL² /3 + 2P/3 /3

2. T = -P(Xt - Xb - St(0) = PL/2

M = -Px/2 + P/2

H(l_c) = -PL/2 - 2P/3 /2

TEOREMA DEI LAVORI VIRTUALI

Li = ¹/_Ei ∫ (χ M_i² + M_Mo) dz = L_iAF + L_iFD

L_iAF = ¹/_Ei ∫₀¹ χ ρ² 4₂ dz = ¹/_Ei χ ρ² 5

L_iFD = ¹/_Ei ∫₀¹ (χ (ρ² 2₂ + ρ² 4 - 2ρ³ 3₂)) dz + ∫₀¹ (ρ(1/2 - ρ - 2)(-2ρ₂ + 2ρ²)) dz

= ¹/_Ei [ (χ ^{ρ2 2₂ 7 + ρ2 42 - 2χ ρ3 3₂})/(³/₃) ]¹ + ∫₀¹ ρ² 2₂ 2 + 2ρ² 2² d₁

= ¹/_Ei [(χ ^ρ(₂ ⁵/₃) + ρ² ρ²/₅) + (-2ρ² 2³/₃ + ρ² 2²/₂+ 2ρ² 3³ - 2ρ² 6²)]

= ¹/_Ei [ (χ ρ^{2 5}/₃) + (-ρ₂³/₃ + ρ² 5 + 2ρ⁵/₂ - 2ρ² 5 - 2ρ⁵)]

= ¹/_Ei [ (χ ρ^{2 5}/₃) - ²/₃ ρ⁵ ]

χ (ρ²⁵/₃ + ρ⁵/₃) - ²/₃ ρ⁵ = 0 → χ (ρ³¹ ρ²⁵) - ²/₃ ρ⁵

→ χ ρ^{5 5}/₃ - ²/₃ ρ⁵⁵ = 0

χ → χ = ²/₃ ²/₄ - ¹/₂ ρ²

M_Mo + M_Lo^x

VINCOLI P₁

PL²

Ɣ_A

ESPLOSO P₁

PL₂

PL²

PL₂

PL²

PL²

3

PL₂

PL²

PL₂