# New PDF release: Finite Elements Part 2 (solid mechanics)

By Victor E Sauoma

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Solve for the reactions, Rt = Kut ∆tt + Kuu ∆u , and overwrite ∆u by Ru −M L/6EI M L/3EI R1 ¾ = R2 L3 /3EI −L/6EI −L/6EI L/3EI 6EI/L2 6EI/L2 −6EI/L2 −6EI/L2 6EI/L2 6EI/L2 12EI/L3 −12EI/L3 −6EI/L2 −6EI/L2 −12EI/L3 12EI/L3 ¿ −M L/6EI M L/3EI 0 0 −M L/6EI M L/3EI = M/L −M/L Cantilivered Beam/Initial Displacement and Concentrated Moment 1. The element stiﬀness matrix is k= −2 ¾ −2 12EI/L3 −3 6EI/L2 −4 −12EI/L3 1 6EI/L2 −3 6EI/L2 4EI/L −6EI/L2 2EI/L −4 −12EI/L3 −6EI/L2 12EI/L3 −6EI/L2 1 ¿ 6EI/L2 2EI/L −6EI/L2 4EI/L −3 2EI/L 6EI/L2 4EI/L −6EI/L2 −4 ¿ −6EI/L2 −12EI/L3 −6EI/L2 12EI/L3 2.

4: Rod subjected to Step Load In this case the element interconnectivity requirements are contained in the assumption that the temperature θ be a continuous function of x and no additional compatibility conditions are applicable. 4. 14) and the initial condition is Example 2-3: Wave Equation, (Bathe 1996) The rod shown in Fig. 4 is initially at rest when a load F (t) is suddenly applied at its free end. 15) where E is the Young’s modulus, ρ the mass density, and A the cross sectional area, c corresponds to the velocity of sound in the elastic medium, and u is the state variable.

We assume the springs to be massless, but the carts have masses mi . 1. 51) 2. 53)  where 3. The equilibrium equation represents a system of ordinary diﬀerential equations of the second order in time. For the solution of these equations it is also necessary to specify the initial conditions for u and u˙ at time t = 0 (u0 and u˙ 0 respectively). 2 Transient Heat Flow Fig. 10 illustrates an idealization of the heat ﬂow inside an electron tube. A ﬁlament is heated to a temperature θf by an electric current; heat is convected from the ﬁlament to the surrounding gas and is radiated to the wall which also receives heat by convection of the gas.