By L. I. Sedov, J. R. M. Radok

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**Extra resources for A course in continuum mechanics, vol. 1: Basic equations and analytical techniques**

**Example text**

3. The parabolic and hyperbolic diﬀerential equations Eq. 12 and Eq. 18 would become elliptic equations if the time-dependent terms were neglected. In this way the initial value problems would be converted to boundary value problems with steady-state solutions. 4. The solution of a boundary value problem depends on the data at all points of the boundary. However, in propagation problem, the solution at an interior point may depend only on the boundary conditions of part of the boundary and the initial conditions over part of the interior domain.

Now we solve for the displacement ∆t = K−1 tt Pt , and overwrite Pt by ∆t ¾ ∆1 θ2 0 0 L3 /3EI 2 = −12EI/L3 L2 /2EI 2 −6EI/L2 L /2EI L/EI 6EI/L 2EI/L −12EI/L3 −6EI/L2 6EI/L2 2EI/L 12EI/L3 6EI/L2 6EI/L2 4EI/L ¿ −P 0 0 0 −P L3 /3EI −P L2 /2EI = 0 0 7. Finally, we solve for the reactions, Ru = Kut ∆tt + Kuu ∆u , and overwrite ∆u by Ru −P L3 /3EI −P L2 /2EI R3 ¾ = R4 ¿ L3 /3EI L2 /2EI L2 /2EI L/EI −12EI/L3 6EI/L2 −6EI/L2 2EI/L −12EI/L3 6EI/L2 12EI/L3 6EI/L2 −6EI/L2 2EI/L 6EI/L2 4EI/L 1 6EI/L2 4EI/L −6EI/L2 2EI/L −4 −12EI/L3 −6EI/L2 12EI/L3 −6EI/L2 2 ¿ 6EI/L2 2EI/L −6EI/L2 4EI/L −P L3 /3EI −P L2 /2EI 0 0 −P L3 /3EI −P L2 /2EI P PL = Simply Supported Beam/End Moment 1.

Establish the equilibrium equations of the problem in terms of these temperatures when the ambient temperatures θ0 and θ4 are known. 6: Slab Subjected to Temperature Boundary Conditions, (Bathe 1996) 1. 25) where q is the total heat ﬂow, A the area, ∆θ the temperature drop in the direction of heat ﬂow, and k the conductance or surface coeﬃcient. 2. The state variables are θ1 , θ2 and θ3 . 3. 26) 4. 28-c) 5. 29) Hydraulic Network In this example, we seek to establish the equations that govern the steady-state pressure and ﬂow distribution in the hydraulic network shown in Fig.