Page 193 - Analysis and Design of Energy Geostructures

166 Analysis and Design of Energy Geostructures

Figure 4.14 An example of axisymmetric problem.
formulations referring to axisymmetric conditions are very useful to analyse problems
involving, for example, single energy piles (cf. Fig. 4.14).
In situations where an axisymmetric formulation can be used, by symmetry the two
components of the displacement in any plane section along the axis of symmetry completely
define the state of strain and hence the state of stress (Zienkiewicz and Taylor, 2005). In par-
ticular, considering a cylindrical coordinate system, any displacement along the radial direc-
tion causes a strain in the circumferential direction that will be associated to a stress different
from zero. The need of considering four strain and stress components in axisymmetric pro-
blems instead of the three involved in plane strain and plane stress problems represents one
crucial difference between the considered alternative formulations of the thermoelastic prob-
lem. One additional difference is that, while the axisymmetric formulation of the thermoe-
lastic problem leads to an equivalent yet simpler analysis of the associated three-dimensional
problem, plane strain and plane stress formulations of the thermoelastic problem lead to an
approximate (simplified) analysis of the associated three-dimensional problem.
The 10 quantities σ rr ; σ zz ; σ θθ ; σ rz ; ε rr ; ε zz ; ε θθ ; ε rz ; u; w satisfy in either case the
following 10 equations:
1. Two equilibrium equations (for the case of no body forces)
8
σ θθ
> @σ rr 1 @σ rz 2 5 0
>
>
<
@r @z r
ð4:64Þ
1 1
> @σ rz @σ zz σ rz
> 5 0
>
:
@r @z r