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AE420/ME471 – Midterm exam
Monday, April 4, 2005 – 9:00am to 9:50pm
Closed notes/closed books/no calculator
Problem 1. FEA of a 2-D thermal problem
Consider the following steady-state thermal problem
2 T
x
2
2 T
y
2
+ Q ( x , y ) = 0 on * ,
where T(x,y) is the unknown temperature field, κ is the thermal conductivity and Q is the
distributed heat source.
We consider the following two boundary conditions
T = T
along "# T
" T
" n
= h T $ T
( f )
along "# q
where
T
is the imposed temperature,
" T
" n
denotes the normal derivative, h is a known
constant (referred to as the film coefficient ) and
T
f
is the temperature of the adjacent fluid
(assumed constant).
The corresponding functional is
$ T
$ x
2
$ T
$ y
2
d 1
1
3 QTd 1
1
T
2 3 T f
T
$ 1 q
Using the variational method, derive the finite element formulation for a generic M - node
“global” element. Clearly indicate the size and content of the vector { d } with the nodal
degrees of freedom. Before you start the derivation, indicate the expected size of the local
stiffness matrix and load vector.
Problem 2. Rayleigh-Ritz and finite element analyses of beam buckling and bending
Consider the beam problem described below.
The beam is of length L , stiffness E , moment of inertia I and is resting on an elastic
foundation of distributed stiffness k. The beam is simply supported at both ends ( x=
and x=L ). It is subjected to a compressive axial force P as shown in the figure, and to a
transverse load q ( x ) (assumed positive in the positive z direction). Let w ( x ) denote the
beam deflection. Assume that E , I , L and P are constant.
The potential energy for this problem can be shown to be
EI
d
2 w
dx
2
2
) P
dw
dx
2
2
dx
0
L
) q ( x ) w dx
0
L
a) If the function w(x) minimizes Π, what ODE does it satisfy?
b) Assuming that there is no transverse loading (i.e., q=0 ), use the Rayleigh-Ritz
method and the simplest trigonometric basis function to obtain an estimate for
the buckling load of the beam (i.e., the critical value of P for which the deflection
w is not zero). Note:
sin
2 m " x
L
( dx
0
L
= cos
2 m " x
L
( dx
0
L
L
, with m being any
non-zero integer.
c) Using the variational (or PMPE) method, derive the finite element formulation for
a 2-node beam element based on the expression of the potential energy given
above. Prior to your derivation, describe the degrees of freedom for this element,
indicate the order of the shape functions, sketch (but do not derive) these shape
functions and indicate the expected size of the stiffness matrix and load vector.
Provide all necessary details on your derivation of the local stiffness matrix [ k ]
and local load vector { r } (you do not need to perform the integrations).
E, I, L
P P
x
z, w ( x )
q ( x )
