By Petr Krysl
Version of a taut cord -- the strategy of Galerkin -- Statics and dynamics examples for the cord version -- Boundary stipulations for the version of a taut cord -- version of warmth conduction -- Galerkin technique for the version of warmth conduction -- Steady-state warmth conduction strategies -- temporary warmth conduction strategies -- increasing the library of point forms -- Discretization errors, errors keep watch over, and convergence -- version of elastodynamics -- Galerkin formula for elastodynamics -- Finite parts for actual 3D difficulties -- interpreting the stresses -- aircraft pressure, aircraft rigidity, and axisymmetric types -- Consistency + balance = convergence
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Extra info for A pragmatic introduction to the finite element method for thermal and stress analysis : with the matlab toolkit SOFEA
We begin with L1 : first we see that the integral should be split into integrals over each element, since the test function N 1 = N2 has different definitions in different elements. L L1 = x3 x2 N 1 N q dx = 1 1 q dx x2 x1 0 N q dx + For element 1 we compute the contribution to L1 as x2 x2 N 1 N2 q dx = qL/4 q dx = x1 x1 and for element 2 we compute the contribution to L1 as x3 x3 N 1 N2 q dx = qL/4 q dx = x2 x2 yielding L1 = qL/2. The load vector component L2 is computed as L L2 = x2 N 2 q dx = x1 0 x3 N 2 N q dx + 2 q dx x2 were the contribution from element 1 is zero, since N to L2 from element 2 is 2 = N3 = 0 in element 1, and the contribution x3 N 2 q dx = qL/4 x2 The load vector is therefore [L] = qL/2 qL/4 The components of the stiffness matrix are computed next.
Also note that the two test functions Nj , j = 1, 2 must not be simultaneously zero at x = L. If that was the case, all effects of the applied force FL would be erased from the formulation, and we couldn’t possibly get a meaningful result. Substituting the test and trial function we have L FL N1 (L) − 0 FL N2 (L) − 0 L ∂N1 ∂N1 P a1 dx − ∂x ∂x ∂N2 ∂N1 P a1 dx − ∂x ∂x The Matlab symbolic algebra reads L 0 0 L ∂N1 ∂N2 P a2 dx = 0 , ∂x ∂x ∂N2 ∂N2 P a2 dx = 0 , ∂x ∂x 26 Thermal and Stress Analysis with the FEM >> syms L P q x FL real q=0; N1=sin(pi*x/L); N2=sin(3/2*pi*x/L); K=int(diff( [N1;N2])*P*diff([N1,N2]),0,L) F=q*int([N1;N2],0,L)+ [FL*subs(N1,x,L);FL*subs(N2,x,L)] a=K\F K = [ (P*pi^2)/(2*L), (9*pi*P)/(5*L)] [ (9*pi*P)/(5*L), (9*P*pi^2)/(8*L)] F = 0 -FL a = (80*FL*L)/(P*pi*(25*pi^2 - 144)) -(200*FL*L)/(9*P*(25*pi^2 - 144)) This yields the approximate solution as shown in the figure below.
Find analytically the natural frequencies and the normal modes of the simply-supported prestressed cable with a uniform mass density.
A pragmatic introduction to the finite element method for thermal and stress analysis : with the matlab toolkit SOFEA by Petr Krysl