CEE504 – Winter 2009 Instructor: Laura N. Lowes
Homework #2 – Due 1/23
Problem #1: Following is the Hellinger-Reissner variational theorem for linear elasticity:
()
00 0
11
22 0
00000
ˆ
(, ) LLLLL
HR
x
xL
Adu
u AE dx dx A dx Adx uTsdx uT A u u stationary
Edx
σεε σσσεσ σ
=
=
Π=− − − + − −−⋅⋅−=
∫∫∫∫∫
following the procedure described in lecture, introduce variations of the independent functions u and
σ, impose the requirement of stationarity, and, identify the governing equations that can be derived from
this theorem. Note that L,σ,u,A,E,Ts, and ˆ
Tare as defined in class; εo represents an initial strain in the
material due to an initial change in temperature, prestress, or other; and u is the displacement
imposed at x=0 (in our example in class this was 0). Note also that this theorem is useful in developing
‘mixed’ finite element formulations in which we want to use different functional forms to approximate the
displacement field, u, and the stress field, σ.
Problem #2: The B.V.P describing response of a cantilever beam subjected to a uniformly distributed
load, assuming Bernoulii-Euler beam bending and thus no shear deformation, is as follows:
22
22
00
0
23
23
0
0; 0
0; 0
xx
x
xL xL
xL xL
ddw
EI q
dx dx
with
dw
wdx
dw dw
EI M EI V
dx dx
θ
==
=
==
==
⎡⎤
−=
⎢⎥
⎣⎦
===
== ==
where E is the elastic modulus of the beam material, I is the beam moment of inertia, w is the deflection
of the beam neutral axis and q is a distributed load.
Using the weighted residual method, derive the weak / variation form of the B.V.P. Note that this form
should include the second derivative of the weight function and the second derivative of the
displacement field. Note also that this will require you to use integration by parts twice. This is the
equation that we will use to derive the FEM formulation for a beam later in the quarter.
Problem #3: For the idealized system shown, compute the exact and approximate displacement fields
as requested below. It will probably be easiest to use matlab, mathematica, or mathcad to do this
problem.
For this system A=2 in2, E=30000 ksi, L=100 in., p(x)=(1e-4)x4 lbf/in., P=1 kip.
a) Solve the strong form of the BVP for u(x):
p(x) P
x = L
x
AE
