9.1 Fundamental Conceptions in Mechanics
523
When ϕ is relatively small, there is tan ϕ ≈ ϕ, and tan ϕ = y
, thus there is
y
= c 1 cos ax + c 2 sin ax
Integrating the above equation gives
y =
c 1
a
sin ax −
c 2
a
cos ax
According to the boundary condition y(0) = 0, we get c 2 = 0, thus there is
y = c sin ax
where, c =
c 1
a
.
Verify that whether the Legendre condition and Jacobi condition are satisfied.
Since
F ϕ ϕ = E I > 0
Obviously the Legendre condition holds.
The Jacobi equation is
E I u
+ Pu = 0 or u
+ a
2 u = 0
Under the boundary condition of u(0) = 0, the solution of the Jacobi equation is
u = d sin ax
When x k =
kπ
a
, where, k = 1, 2, . . ., the value of the function u vanishes. If
L ≥
π
a
, then the Jacobi condition is completely satisfied, therefore
P ≥
π
2
L 2 E I
The minimum of the Euler critical compressive force is
P min =
π
2
L 2 E I
The deflection curve equation of the bar is
y = c sin
π x
L