AE420/ME471 – Introduction to the Finite Element Method
Final Exam – Spring 2007
3 hours – Closed notes/closed books/no calculator
Total: 35 points
Problem 1. Theory (7 points)
1. What are the basic idea and the motivation behind the introduction of mapped elements?
2. What is the difference between Euler and Mindlin beam theories? Which beam element locks
and why? What can be done to solve the locking problem and why does it work?
3. Show that the backward difference scheme is always stable for first-order (thermal) transient
problems.
Problem 2. Transient plane stress problem (13 points)
The principle of virtual work (PVW) for a transient plane stress problem is given by
!" #
{ }
d$
$
% +
!
u
&
!!
u
{ }
d$
$
%=
!
u F
{ }
d$
$
% +
!
u T
{ }
d'T
'T
%, (
!
u
{ }
,
where
!
denotes the material density,
u=u x,y,t
( )
v x,y,t
( )
is the (transient) displacement
vector,
T=TxTy
is the (possibly transient) applied traction vector (along the portion
!
T
of
the boundary), and
F=FxFy
is the (possibly transient) body force vector.
The strain
!
{ }
and stress
!
{ }
vectors are defined by
!
{ }
=
"
xx
"
yy
!
xy
#
$
%
&
%
'
(
%
)
%
=
*u/*x
*v/*y
*u/*y+*v/*x
#
$
%
&
%
'
(
%
)
%
,
!
{ }
=
!
xx
!
yy
!
xy
"
#
$
%
$
&
'
$
(
$
=D
[ ]
)
{ }
,
where
D
[ ]
is the symmetric 3*3 material stiffness matrix. A superposed dot denotes derivative
with respect to time,
!
u
and
!"
respectively denote the virtual displacement and strain
vectors.
a) Using the PVW approach, derive the semi-discrete finite element formulation for this
problem for a M-node transient plane stress element. Indicate clearly the content of the
vector with the local degrees of freedom. Prior to the derivation of the local stiffness [k]
and mass [m] matrices and of the local load vector {r}, indicate their expected size. Make
sure to provide the complete expression of all quantities used in your derivation.
b) Give the expression of [k], [m] and {r} for the 9-node isoparametric 2D quadrilateral
plane stress element. Write the expression of the shape functions associated with each
node and derive all the quantities entering the expression of [k], [m] and {r} (you do not
need to actually perform the integrations!).
Problem 3. Transient axisymmetric membrane problem (9 points)
The transient transverse deflection
w r,t
( )
of an annular shape membrane of inner and outer radii
a and b, respectively, subjected to a radially varying distributed transient transverse load
q r,t
( )
and supported by an elastic foundation of (constant) distributed stiffness
ks
is described by
