Page 181 - Analysis and Design of Energy Geostructures
P. 181
154 Analysis and Design of Energy Geostructures
The indefinite equilibrium equations imply that if an element is in equilibrium, when
introducing body forces the sum of the forces in each direction must be zero. In com-
pact form, this reads
r σ ij 1 ρb i 5 0 ð4:38Þ
where r is the divergence operator, σ ij is the total stress tensor, ρ is the density of the
material and b i is the vector of the body forces (per unit mass). At the surface, surface
forces substitute body forces and the corresponding equations are indicated as bound-
ary conditions (or boundary equilibrium equations, cf. Section 4.7).
Eq. (4.38) is as follows in relevant coordinate systems.
• Cartesian coordinates x, y, z:
1 @σ xy 1 @σ xz 1 ρX 5 0
@σ xx
@x @y @z
1 @σ yy 1 @σ yz 1 ρY 5 0 ð4:39Þ
@σ yx
@x @y @z
1 @σ zy 1 @σ zz 1 ρZ 5 0
@σ zx
@x @y @z
where X ; Y and Z are the components of the body forces in the x; y and z
directions, respectively.
• Cylindrical coordinates r, θ, z:
1 @σ rθ σ rr 2σ θθ
1 1 @σ rz 1 1 ρR 5 0
@σ rr
r @θ
@r @z r
1 1 @σ θz 1 @σ zz 1 σ rz 1 ρZ 5 0
@σ rz
r @θ ð4:40Þ
@r @z r
@σ rθ 1 @σ θθ 2σ rθ
1 1 @σ θz 1 1 ρΘ 5 0
r @θ
@r @z r
where the body force components in the r; z and θ directions are denoted by
R ; Z and Θ .
• Spherical coordinates r, θ, φ:
1 @σ rθ 1 @σ rφ 1
1 1 1 ½2σ rr 2 σ θθ 2 σ φφ 1 σ rθ cotθ 1 ρR 5 0
@σ rr
r @θ rsinθ @φ
@r r
@σ rθ 1 @σ θθ 1 @σ θφ 1
1 1 1 ½ 2σ θθ 2 σ φφ cotθ 1 3σ rθ 1 ρΘ 5 0
r @θ rsinθ @φ ð4:41Þ
@r r
@σ rφ 1 @σ θφ 1 @σ φφ 1
1 1 1 ½3σ rφ 1 2σ θφ cotθ 1 ρΦ 5 0
r @θ rsinθ @φ
@r r
