AE420/ME471 – Spring 2007 – First Midterm
Friday, February 23, 2007
50 minutes – Closed notes/closed books/no calculator
Question 1: Rayleigh-Ritz Method (7 points)
Consider the simple beam buckling problem described in Figure 1.
Figure 1
The beam is of length L, stiffness E and moment of inertia I. It is simply supported at both ends, x=0 and
x=L. It is subjected to a compressive axial force P applied at both ends and to a distributed transverse load
q(x) (as shown in Figure 1). Note: we only solve for the transverse deflection w, not for the axial
displacement. The boundary conditions for this problem are simply
w0
( )
=!!
w0
( )
=w L
( )
=!!
w L
( )
=0
.
The potential energy for this problem can be shown to be
!=1
2EI d2w
dx2
"
#
$%
&
'
2
(1
2Pdw
dx
"
#
$%
&
'
2
(qw
)
*
+
+
,
-
.
.dx
0
L
/
.
a) Denoting the approximate deflection solution as
!
w x
( )
=aifix
( )
i=1
N
!
,
where the
ai
are the unknown coefficients and
fi
are the basis functions, derive the general expression of the
linear system
K
[ ]
A
{ }
=R
{ }
used to solve the N-component vector
A=a1a2... aN
. Write the matrix [K] and vector {R} both in
“matrix/vector” (i.e., [K]=…, {R}=…) and component (i.e.,
Kij =
…,
Ri=
…) forms. What are the
conditions on the basis functions to make them admissible?
b) Assuming N=1, use the simplest polynomial basis function to obtain an approximation for the buckling
load for this problem (i.e., the critical value of P for which the deflection w is not zero for zero transverse
load
q x
( )
=0
).
Question 2: Assembling (2 points)
Consider the simple 2-D truss structure shown in
Figure 2. It is composed of 4 nodes and 5
elements.
The connectivity table linking nodes to elements
is as follows:
Element #
Node a
Node b
1
1
3
2
4
2
3
3
2
4
3
4
5
4
1
E, I, L
P
q(x)
Figure 2
P
