For the applied loads:
distributed axial force f(x), distributed transverse load q(x), horizontal point load P at x=L,
transverse point load F at x=L, and bending moment Mo (counterclockwise) at x=L the potential
is given by
00 0 0 0 ( ) 0
0 ( ) () ()
L
x L
dw
V fu qw dx Pu L Fw L M dx =
− =−
∫
The total potential energy functional of the beam is given by 0 0 ∏ = (, ) uw U V
Determine the governing equations (Euler equations) and boundary conditions (35 points).
Problem #3: Determine the Euler equations and the natural boundary conditions associated with
the functional
2 2 2 2 2 22 2
2
2 2 22
1 ( ) 2 2(1 ) 2 2
D w w ww w
w dxdy kw dxdy qwdxdy