Lezione 7 di Costruzione di macchine

Solidi assialsimmetrici

Basic hypotheses

Basic hypotheses: full axisymmetry: generali.

• Geometrical.
• Loads.
• Material (it has to be also linear elastic).

State of stress:

• Plane stress (disks).
• Plane strain (tubes).

Coordinate system: cylindrical.

• z axial coordinate.
• r radial coordinate.
• theta circumferential coordinate.

Reference system: sdr del pezzo.

• z axial direction.
• r radial direction.
• c circumferential direction.

Trascuro lo sforzo in una direzione (tensione piana).

co 1 riga no 1 colonna.

If the axisymmetric hypotheses hold, the state of stress only depends on the radial coordinate:

sigma(r, theta, z) = sigma(r)

The state of stress is triaxial non più 1D!

• Radial stress sigma_r = sigma_r(r) radiale.
• Circumferential stress sigma_c = sigma_c(r) tangenziale.
• Axial stress sigma_z = sigma_z(r) assiale.

Radial sigma_r and circumferential sigma_c stresses are independent of end constraints vincoli.

Axial stress sigma_z depends on end constraints.

Condizioni generali e sistema di riferimento

Solidi assialsimmetrici.

Basic hypotheses: full axisymmetry: -> conditions generali.

• Geometrical.
• Loads.
• Material (it has to be also linear elastic).

State of stress:

• Plane stress (disks).
• Plane strain (tubes).

Coordinate system: cylindrical.

• z axial coordinate.
• r radial coordinate.
• theta circumferential coordinate.

Reference system: -> sdr del pezzo.

• z axial direction.
• r radial direction.
• c circumferential direction.

Trascuro lo sforzo -> in una direzione (tensione piana).

no 1 riga no 1 colonna.

If the axisymmetric hypotheses hold, the state of stress only depends on the radial coordinate:

sigma(r, theta, z) = sigma(r)

The state of stress is triaxial -> non più 1D!

• Radial stress sigma_r = sigma_r(r) -> radiale.
• Circumferential stress sigma_c = sigma_c(r) -> tangenziale.
• Axial stress sigma_z = sigma_z(r) -> assiale.

Radial sigma_r and circumferential sigma_c stresses are independent of end constraints -> vincoli.

Axial stress sigma_z depends on end constraints.

Classificazioni

3 classificazioni --> in base ai rapporti di certi numeri.

in base a questi rapporti si possono chiamare il componente!

Discs and rings:

• Radial stress σ_r and circumferential stress σ_c are principal stresses.
• Axial stress σ_z = 0 -> state of plane stress.

Tubes:

• Radial stress σ_r, circumferential stress σ_c, and axial stress σ_z are the three principal stresses.
• Axial strain ε_z = 0 -> state of plane strain, ε_z = ε₀ constant -> generalized plane strain state.
• Radial stress σ_r, circumferential stress σ_c depend on the internal and external pressure p_i and p_e.
• Axial stress σ_z depends only on end constraints.

Equations: equilibrium, strain compatibility, material constitutive (linear elasticity).

Top/Bottom head shells are not examined, as well as next the welded connection with the cylindrical shell:

• Heads are studied with the theory of plates and shells.
• Welded connections generate border effects (stress concentrations).

According to the ratio between the radial thickness r and the internal radius, there is a distinction:

1. Thick tube: t / r_i > 0.1 -> σ_r(r) from Lamé theory.
2. Thin tube: t / r_i ≤ 0.1 -> σ_r from Boyle-Mariotte theory.

I° caso

si considera la componente verticale perché quella azzerata non compare le sc! si fa equivalere sul solido.

Thin tube with Internal Pressure.

Invece.

The internal border is the most stressed:

Radial equilibrium: _c(tL) = p_i(D_iL) -> σ_c = p_i D_i / 2t = const.

Radiale.

sece su cui poggiano le pareti del tubo.

una specie di reazione media nell'area (t·L)².

Constitutive equations

Constitutive equations of the (elastic) material:

ε_c = 1/E [σ_c - v(σ_r + σ_z)] + αΔT

ε_r = 1/E [σ_r - v(σ_c + σ_z)] + αΔT

ε_z = 1/E [σ_z - v(σ_r + σ_c)] + αΔT

Mechanical strain.

Thermal expansion.

α, Coefficient of Thermal Expansion (CTE, 1/°C or °C^-1)

uniform in quesidirezione, da considerare.

Compatibility equation

The relation between the radial displacements u and the circumferential strain ε_c (called the compatibility equation) can be easily obtained by considering the length of a generic fiber at radius r before and after the radial displacement:

ε_c = (l_f - l_i) / l_i = (2π(r + u) - 2πr) / 2πr = u / r

Therefore, the radial displacement can be computed as:

u = rε_c

The radial displacement of interest is at the internal radius.

Axial stress σ_z can be evaluated on the basis of the end conditions.

Axial equilibrium

Axial equilibrium -

Axial equilibrium -: 

From the constitutive equation:

eps_c = 1/E [sigma_c - v(sigma_r + sigma_z)] + alpha Delta T

eps_c = p_i/E [D_i/2t (1 - v/2) + v] + alpha DeltaT -> u_i = D_i/2 [p_i/E [D_i/2t (1 - v/2) + v] + alpha DeltaT]

Libero di muoversi al suo interno ma con due tappi ai lati > sigma_z costante in pressione (come prova di trazione).

Axial equilibrium -:

The tube is unloaded.

sigma_z = 0

ora tubo aperto.

No carico!

From the constitutive equation:

eps_c = 1/E [sigma_c - v(sigma_r + sigma_z)] + alpha DeltaT

eps_c = p_i/E [D_i/2t + v] + alpha DeltaT -> u_i = D_i/2 [p_i/E [D_i/2t + v] + alpha DeltaT]

estremi si muovono verticali, no assiali no def. assiale!

Axial equilibrium -:

The axial strain is prevented:

eps_z = 0

From the constitutive equation:

eps_z = 1/E [sigma_z - v(sigma_r + sigma_c)] + alpha DeltaT = 0

The axial stress results:

sigma_z = v(sigma_c + sigma_r) - Ealpha DeltaT -> sigma_z = v p_i [D_i/2t - 1] - Ealpha DeltaT

From the constitutive equation:

eps_c = 1/E (1 + v) [D_i/2t (1 - v) + v] + alpha DeltaT (1 + v) -> u_i = D_i/2 [p_i/E (1 + v) [D_i/2t (1 - v) + v] + alpha DeltaT (1 + v)]

Case, axial stress, axial strain, circumferential strain, radial displacement