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static_beam-finite_difference/README.md

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Numerically integrate Euler-Bernoulli equation

$EI\frac{d^4w}{dx^4}=0$

Boundary Conditions:

  1. $w(0)=0$

  2. $w'(0)=0$

  3. $w''(L)=0$

  4. $EIw'''(L) = -F$

Using constants

  • L=400 mm
  • E=200e3 MPa
  • t=3 mm
  • w=12 mm

Demonstrate qualitative convergence between 4, 24, and 44 nodes. The beam shape looks similar between each solution

Demonstrate quantitative convergence of tip displacement under given load.

The relative error is calculated based upon finite-difference calculations

$e_{rel}=\frac{\delta_{new}-\delta_{old}}{\delta_{new}}$

The absolute error is compared to the analytical solution of $EIw''''=0$ for a cantilever beam

$\delta_{an}=\frac{FL^3}{3EI}$

$e_{abs}=\frac{\delta_{new}-\delta_{an}}{\delta_{an}}$