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1D Bar FEM Explorer

Interactive tool · Finite element method

1D Bar FEM Explorer

About this tool

An axially loaded bar, fixed at one end. Change the material, geometry, loads and mesh, and compare the finite element displacement and stress with the exact solution.

Alatan interaktif · Finite element method

1D Bar FEM Explorer

Tentang alatan ini

Bar yang dibebankan secara paksi, ditetapkan pada satu hujung. Ubah bahan, geometri, beban dan mesh, dan bandingkan sesaran dan tegasan finite element dengan exact solution.

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How it works

A bar fixed at x = 0 carries a tip load P and a uniform distributed load q, with an area that varies linearly from A₀ at the support to A(L) at the tip. The governing equation and its exact solution are

d/dx ( E A(x) du/dx ) + q = 0     u(0) = 0     E A du/dx |x=L = P     N(x) = P + q (L − x)

Each element contributes a stiffness matrix and a consistent load vector,

ke = ∫ BT E A B dx     fe = ∫ NT q dx     linear, uniform bar: ke = (EA / Le) [1 −1; −1 1], fe = (q Le / 2) [1; 1]

The element matrices are assembled into K u = f, the fixed node is removed, and the stress follows from σ = E du/dx. Linear elements give a constant stress in each element; quadratic elements give a linearly varying stress.

  • Tip load only, uniform bar: the axial force is constant, so linear elements are exact everywhere. This is the case of the original app, umax = PL / EA.
  • Distributed load: linear elements still give exact nodal displacements (a property of one-dimensional problems with constant EA), but the stress inside each element is only an average of the exact linear variation.
  • Tapered bar: the exact displacement is logarithmic, so the nodal values are no longer exact, and the error falls with the mesh size.
Things to try
  1. Keep the default tip load and change the number of elements. The displacement and stress stay exact.
  2. Add q = 100 kN/m. Compare the stress plot with the exact line, then switch to quadratic elements.
  3. Set the taper to 0.25 and run through n = 1, 2, 4, 8. Read the convergence plot: linear elements converge at rate 1 in the energy norm, quadratic elements at rate 2.
  4. Make P negative to put the bar in compression and watch the colour change.
  5. Open the tables to compare the nodal values with the exact solution, and copy the Python code to run the same model yourself.
Bagaimana ia berfungsi

Satu bar yang ditetapkan pada x = 0 menanggung beban hujung P dan beban teragih seragam q, dengan luas yang berubah secara linear dari A₀ di support ke A(L) di hujung. Persamaan menakluk dan exact solution-nya ialah

d/dx ( E A(x) du/dx ) + q = 0     u(0) = 0     E A du/dx |x=L = P     N(x) = P + q (L − x)

Setiap element menyumbang stiffness matrix dan consistent load vector,

ke = ∫ BT E A B dx     fe = ∫ NT q dx     linear, bar seragam: ke = (EA / Le) [1 −1; −1 1], fe = (q Le / 2) [1; 1]

Matriks element dipasang menjadi K u = f, nod yang ditetapkan dikeluarkan, dan tegasan diperoleh daripada σ = E du/dx. Element linear memberi tegasan malar dalam setiap element; element kuadratik memberi tegasan yang berubah secara linear.

  • Beban hujung sahaja, bar seragam: daya paksi adalah malar, jadi element linear exact di semua tempat. Inilah kes app asal, umax = PL / EA.
  • Beban teragih: element linear masih memberi sesaran nodal yang exact (sifat masalah satu dimensi dengan EA malar), tetapi tegasan dalam setiap element hanyalah purata bagi perubahan linear yang exact.
  • Bar tirus: sesaran exact adalah logaritma, jadi nilai nodal tidak lagi exact, dan ralat berkurang mengikut saiz mesh.
Cubaan untuk anda
  1. Kekalkan beban hujung lalai dan ubah bilangan element. Sesaran dan tegasan kekal exact.
  2. Tambah q = 100 kN/m. Bandingkan plot tegasan dengan garis exact, kemudian tukar ke element kuadratik.
  3. Tetapkan tirus kepada 0.25 dan cuba n = 1, 2, 4, 8. Baca plot convergence: element linear menumpu pada kadar 1 dalam energy norm, element kuadratik pada kadar 2.
  4. Jadikan P negatif supaya bar dalam mampatan dan lihat warnanya berubah.
  5. Buka jadual untuk membandingkan nilai nodal dengan exact solution, dan salin kod Python untuk menjalankan model yang sama sendiri.