ABSTRACT
The gradient operator ∇ is a column matrix. In rectangular Cartesian coordinates,
∇ = ∂∂ ∂
∂ ∂
∂
x y z .
(7.3)
In cylindrical coordinates,
∇ = ∂
∂ ∂
∂ ∂
∂
r r z
1 θ
. (7.4)
The equivalent virtual work formula is generated by multiplying 7.2 by arbitrary functions T (i.e., a virtual temperature variation δT T= ), integrating over the volume, and applying integration by parts to obtain
ρcT T t
V T T V T T t
∂ ∂ + ∇ ∇ +
∂ ∂ = − +∫ ∫ ∫ ∫d d d dT T( ) κ β ε0 T r V
ρ d∫ ,
(7.5)
where
q = –nTκ∇T (7.6)
is the heat flux into the body across the surface with outward normal n. After dividing the volume into finite elements, we introduce shape functions for
each element:
u = ND, ε = AD, T = ST, ∇T = BT, (7.7)
where T is the column matrix of nodal temperatures. Using the same shape functions to generate the virtual temperature variations yields the finite element form of the virtual work:
C T K T Q C Dtt tt tu + = − , (7.8)
where
K B Btt
dκ , (7.9)
C S Stt
d , (7.10)
C S Atu
dβ , (7.11)
Q S S= − +∫∑ ∫∑T elements
d dq A r V S V
ρ .
