Page 191 - Analysis and Design of Energy Geostructures

164 Analysis and Design of Energy Geostructures

Figure 4.13 Typical configuration for (A) (C) plane strain and (D) plane stress problems.

of their thickness (cf. Fig. 4.13D). The mathematical formulation corresponding to
plane stress conditions involves

σ zz 5 σ xz 5 σ yz 5 0 ð4:56Þ

Especially the plane strain formulation has been widely used to analyse conven-
tional problems involving geostructures such as walls, tunnels, slabs and piles.
However, this formulation can only be used for preliminary analyses of energy
geostructures such as energy walls, energy tunnels, energy slabs and energy piles. The
reason for this is that the effect of temperature on the deformation of the geostructure
that would conventionally be neglected in the third direction plays in this case a
remarkable role for the understanding of its mechanical behaviour.
The eight quantities σ xx ; σ yy ; σ xy ; ε xx ; ε yy ; ε xy ; u; v satisfy in either case the following
eight equations:

1. Two equilibrium equations (for the case of no body forces)
8
> @σ xx 1 @σ xy 5 0
>
>
<
@x @y
ð4:57Þ
1
> @σ xy @σ yy
> 5 0
>
:
@x @y
2. Three stress strain relations
8
ε xx 5 1
ð
> σ xx 2 νσ yy 2 α T 2 T 0 Þ
>
>
>
E
>
>
>
< 1
ε yy 5 σ yy 2 νσ xx 2 α T 2 T 0 Þ ð4:58Þ
ð
E
>
>
>
>
ε xy 5
> σ xy
>
>
:
2G