{"id":1673,"date":"2026-10-19T21:58:31","date_gmt":"2026-10-19T13:58:31","guid":{"rendered":"https:\/\/people.utm.my\/arazin\/?page_id=1673"},"modified":"2026-11-07T20:53:39","modified_gmt":"2026-11-07T12:53:39","slug":"1d-bar-fem-explorer","status":"publish","type":"page","link":"https:\/\/people.utm.my\/arazin\/1d-bar-fem-explorer\/","title":{"rendered":"1D Bar FEM Explorer"},"content":{"rendered":"
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Interactive tool · Finite element method<\/p>\n

1D Bar FEM Explorer<\/h1>\n
\nAbout this tool<\/summary>\n

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.<\/p>\n<\/details>\n<\/div>\n

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Alatan interaktif · Finite element method<\/p>\n

1D Bar FEM Explorer<\/h1>\n
\nTentang alatan ini<\/summary>\n

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.<\/p>\n<\/details>\n<\/div>\n

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Loading the interactive tool…<\/span>Memuatkan alatan interaktif…<\/span><\/p>\n<\/div>\n

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\nHow it works<\/summary>\n
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A bar fixed at x = 0 carries a tip load P and a uniform distributed load q, with an area that varies linearly from A\u2080 at the support to A(L) at the tip. The governing equation and its exact solution are<\/p>\n

d\/dx ( E A(x) du\/dx ) + q = 0     u(0) = 0     E A du\/dx |x=L<\/sub> = P     N(x) = P + q (L − x)<\/div>\n

Each element contributes a stiffness matrix and a consistent load vector,<\/p>\n

ke<\/sub> = ∫ BT<\/sup> E A B dx     fe<\/sub> = ∫ NT<\/sup> q dx     linear, uniform bar: ke<\/sub> = (EA \/ Le<\/sub>) [1 −1; −1 1], fe<\/sub> = (q Le<\/sub> \/ 2) [1; 1]<\/div>\n

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.<\/p>\n