Transient conduction with finite element analysis

[rgh107]

Homework Statement

Hi, I've got a civ engineering problem relating to conductive heat flow using finite element analysis.
I'm investigating the dissipation of heat from square and circular pile foundations. I'm applying a thermal load to the centre of the concrete pile which induces heat flow through the surrounding soil [homogeneous material]. At the end of the test cycle I am able to plot a temperature gradient in the horizontal direction through the soil. I was wondering, is there a way by which I can approximate the heat flow into the soil over the test period from the temperature gradient in terms of heat flux density? It should be noted that the finite element model is dependent on thermal conductivity, heat capacity and density.

The key problem I have is that I'm applying cyclic loading by which the magnitude of the applied thermal load is constantly varied over the course of the test period.

It should be noted that due to the nature of the problem, I am seeking only approximate solutions as a means of conducting a simple comparison with different model specifications.

Any help would be most appreciated. . .

Homework Equations

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The Attempt at a Solution

I've attempted obtaining solutions using the basic heat equation. However, this assumed steady-state conditions.

 

[BigJDubs]

I'm struggling to find a solution which accurately accounts for the cyclic loading conditions. I was wondering if anyone could suggest some approaches? I'm aware that finite element analysis is typically used to solve transient heat equations, but I'm unsure how to apply it in this case. I'm also aware of the existence of Laplace transforms which may be used to solve the problem, but again lack the knowledge to do this. Any help would be much appreciated.