Lecture 1 Notes: Equilibrium
We consider a body (solid or fluid) and define the following quantities:
= Surface on which displacements, velocities are
Su prescribed
Sf = Surface of applied forces or heat fluxes
S
Forces per unit surface
f f =
B
f = Forces
Lecture 3 Notes: Surface Displacement
Su= Surface on which displacements are prescribed
Sf= Surface on which loads are applied
Su Sf= S ; Sf Su=
Given the system geometry (V, Su, Sf ), loads (fB , fSf ), and material laws, we calculate:
Displacements u,
Lecture 4 Notes:
Node Mathematics
N is the number of nodes (3N = n) andHis the displacement interpolation matrix. For the
moment, letsassume Su = 0. We use
Then, we obtain
uT =
u1 u2 u3 . . . un
(m)
u
(m) = B
61
6nn1
(2)
We also assume
(m
(m)
u)
= H u
Lecture 6 Notes: Variational Formulas
Discretization of the variational formulation leads to the following equilibrium statement:
t+tF= t+tR
t+t
(1)
F = nodal forces corresponding to element stresses at time
t+
t
t+t
R = external loads applied at the node
Lecture 5 Notes: Static Analysis
KU = R
K = mK
; K
=
(m
(m)
)
R = R B + RS
B =m
B
R
V(m)
V
B
(1)
B
C
B
(m) (m (m
T
)
)
d
V
(m)
(m)
H(m)T fB(m)dV
R(m)
m S
;R(m) =
S
i
(m)
Sfi(m)
f
i(m)
T
HSf
RS = R(m)
;R(m)
u(m)
=
i(m)
fSf
dSi(m)
H(m)U
=
(m)
= B(m)U
Note
Lecture 9 Notes: DOF
u2
u
1
u3
4
u
should be positive, and should remain positive.
Our rule: Remove clamps one at a time, in the order we would perform Gauss elimination. If
there is a clamp seeing no more stiness after having removed some clamp(
Lecture 10 Notes: Wave Intensity
C = Lw (C depends on material properties)
tw
For this system, C is the wave speed (given), Lw is the critical wavelength to be represented, tw is
the total time for this wave to travel past a point, Le is the eective lengt
Lecture 7 Notes Virtual Temperature
We obtain the result
k
+q =0
2
x
x
B
2
k
L=
S
q
in V
=0
L ,
x=0
Principle of Virtual Temperatures
Clearly:
2
k
2
x
+q
2
V
k
x
2
+q
B
dV = A
0
=0
k x
2
Hence,
L
0
2
k x
2
+q B dx= 0
2
(A)
2
L
B
+qB
dx= 0
L
kx
R=
Lecture 2 Notes: Constitutive Relations
The fundamental conditions to be satisfied are:
I. Equilibrium: in solids, F = ma; in fluids, conservation of momentum
II. Compatibility: continuity and boundary conditions
III. Constitutive relations: Stress/str