fvr.structure module

Structure Beam and Tube objects.

class Beam[source]

Bases: object

Euler-Bernoulli beam.

__init__(E, I, A, L, rho)[source]
Parameters:

  • E (float) – Elastic modulus / Young’s modulus

  • I (float) – Second moment of area of the beam’s cross-section

  • A (float) – Cross-section

  • L (float) – Length

  • rho (float) – Density

\[I = \frac{h^3}{12}\]
eigenfrequency(n, support='fixed-free')[source]

Natural frequencies of the beam.

Parameters:

  • n (int) – Mode number (1 for first mode, …)

  • support (str) – only ‘fixed-free’ atm.

Returns:

n-th natural frequencies of vibration !!! Currently only the first four frequencies can be calculated.

Return type:

float

Dynamic beam equation, the Euler-Lagrange equation, of an Euler-Bernoulli beam. The governing differential equation of motion, with \(\mu\) the mass per unit length and \(\delta\) the viscous damping per unit length.

\[\frac{\partial^2}{\partial x^2} \left( EI(x) \frac{\partial^2 y}{\partial x^2} \right) + \mu(x)\frac{\partial^2 y}{\partial t^2} + \delta (x)\frac{\partial y}{\partial t} = f(x, t)\]

Consider the undamped mode in bending vibration of the beam with uniform sectional property. The natural frequencies and mode shapes are obtained considering the homogeneous solution of the beam vibration equation. The Ansatz is

\[y(x, t) = \phi(x)\sin(\omega t)\]

with \(\phi\) being mode shape functions and \(\omega\) the angular natural frequency. Substituting gives

\[\frac{\mathrm{d}^4\phi}{\mathrm{d}x^4} - a^4 \phi(x) = 0\]

with \(a^4:=\mu\omega^2/(EI)\) or

\[a_n := \left(\frac{\mu\,\omega_n^2}{E\,I}\right)^{1/4}\]

The general solution is

\[\phi(x) = A\sin{ax} + B\cos{ax} + C\sinh{ax} + D\cosh{ax}\]

with \(A,B,C,D\) being integration constants and defined by the coundary conditions.

fixed-free support (free vibration of a cantilever beam). Characteristic equations:

\[\cos{aL}\,\cosh{aL}+1=0\]

\(a_n\) is a numerically solved value: \(a_1 L = 0.596864...\pi\), \(a_2 L = 1.49418...\pi\), \(a_3 L = 2.50025...\pi\), \(a_4 L = 3.49999...\pi\), …

\[\omega_n = a_n^2\sqrt{\frac{E\,I}{\mu}} \quad,\quad f_n = \frac{a_n^2}{2\pi}\sqrt{\frac{E\,I}{\mu}}\]
property V: float
property m: float
property mu: float

Mass per unit length (or the product of density and cross-section)

class Plate[source]

Bases: object

Thin circular plate uniformly loaded with external pressure.

Important

Can be used as long as the corresponding stress does not exeed the proportional limit of the material.

References

  • Timoshenko, Stephen P., and S. Woinowsky-Krieger. 1959. Theory of Plates and Shells. 2nd ed. New York: McGraw-Hill Book.

__init__(E, nu, ra, h)[source]
Parameters:

  • E (float) – Elastic modulus / Young’s modulus

  • nu (float) – Poisson’s ratio

  • ra (float) – radius

  • h (float) – thickness

deflection(q, support='clamped')[source]

Central deflection of a clamped supported circular plate.

Parameters:

  • q – external pressure

  • support (str) – clamped, simple

Return type:

float | None

See also

deflection_clamped() and deflection_simple()

deflection_clamped(q)[source]

Central deflection of a clamped supported circular plate.

Parameters:

q – external pressure

Return type:

float

\[w_\text{max} = \frac{q \, r_\text{a}^4}{64D}\]

References

  • Timoshenko, Stephen P., and S. Woinowsky-Krieger. 1959. Theory of Plates and Shells. 2nd ed. New York: McGraw-Hill Book. p. 55.

deflection_simple(q)[source]

Central deflection of a simple/pinned supported circular plate.

Parameters:

q – external pressure

Return type:

float

\[w_\text{max} = \frac{q \, r_\text{a}^4}{64D} \frac{5+\nu}{1+\nu}\]

References

  • Timoshenko, Stephen P., and S. Woinowsky-Krieger. 1959. Theory of Plates and Shells. 2nd ed. New York: McGraw-Hill Book. p. 57.

property D: float

Flexural rigidity of the plate.

\[D = \frac{E\,h^3}{12(1-\nu^2)}\]
class TubeBuckling[source]

Bases: object

Long thin circular tube uniformly loaded with external pressure.

Elemental ring of unit width (h)

Important

Can be used as long as the corresponding compressive stress does not exeed the proportional limit of the material.

References

  • Timoshenko, Stephen P., and James M. Gere. 1961. Theory of Elastic Stability. 2nd ed. New York: McGraw-Hill Book. ch. 7.

__init__(E, nu, r, *, h=None, q=None, s=None)[source]
Parameters:

  • E (float) – Young’s modulus

  • nu (float) – Poisson’s ratio

  • r (float) – mean radius (\(r_\text{a}\) + \(r_\text{i}\))/2

  • h (float | None) – thickness

  • s (float | None) – internal stress

  • q (float | None) – external pressure

force()[source]

Critical buckling value of the compressive force.

A long circular tube uniformly compressed by external pressure.

\[f_\text{cr} = \frac{E h^3}{4 \, (1 - \nu^2) \, r^2}\]

References

  • Timoshenko, Stephen P., and James M. Gere. 1961. Theory of Elastic Stability. 2nd ed. New York: McGraw-Hill Book. p. 289.

Return type:

float

pressure()[source]

Critical buckling value of the compressive pressure.

A long circular tube uniformly compressed by external pressure.

\[q_\text{cr} = f_\text{cr}/r\]
\[q_\text{cr} = \frac{E}{4 \, (1 - \nu^2)} \left(\frac{h}{r}\right)^3\]

References

  • Timoshenko, Stephen P., and James M. Gere. 1961. Theory of Elastic Stability. 2nd ed. New York: McGraw-Hill Book. p. 289.

Return type:

float

stress()[source]

Critical buckling stress of a long thin circular tube uniformly compressed by pressure.

\[\sigma_\text{cr} = f_\text{cr}/h\]
\[\sigma_\text{cr} = \frac{E}{1 - \nu^2} \left(\frac{h}{2r}\right)^2\]

References

  • Timoshenko, Stephen P., and James M. Gere. 1961. Theory of Elastic Stability. 2nd ed. New York: McGraw-Hill Book. p. 293.

Return type:

float

thickness()[source]

Critical buckling thickness of a long thin circular tube uniformly compressed by external pressure.

Returns:

  • Thickness regarding the internal stress, if stress is given

  • Thickness regarding the external pressure, if pressure is given

  • Otherwise given thickness

  • Or None

Return type:

float | None

See also

buckling_thickness_pressure() and buckling_thickness_stress()

thickness_pressure()[source]

Critical buckling thickness of a long thin circular tube uniformly compressed by external pressure.

Thickness regarding the external pressure

\[h_\text{cr,q} = \left(q_\text{cr} \, 4 \frac{1 - \nu^2}{E}\right)^\frac{1}{3} {r}\]

References

  • Timoshenko, Stephen P., and James M. Gere. 1961. Theory of Elastic Stability. 2nd ed. New York: McGraw-Hill Book. p. 293.

Return type:

float

thickness_stress()[source]

Critical buckling thickness of a long thin circular tube uniformly compressed by external pressure.

Thickness regarding the internal stress

\[h_{\text{cr,}\sigma} = \sqrt{\sigma_\text{cr} \frac{1 - \nu^2}{E}} {2r}\]

References

  • Timoshenko, Stephen P., and James M. Gere. 1961. Theory of Elastic Stability. 2nd ed. New York: McGraw-Hill Book. p. 293.

Return type:

float