9.2. Euler’s formula for the critical force
Let us consider the bar in a critical state when the compressive force achieves the critical value i.e. accept that the bar is slightly bent (Fig. 9.3). If the moments of inertia about two principal central axes of the cross section are not equal, then the elastic buckling will be in the minimum rigid plane i.e. the cross section will be about the axis whose inertia moment has the minimum. It is easy to see while compressing a flexible ruler.
To study the elastic buckling and to determine the critical force let’s use the approximate differential equation of the deflection curve of the bent beam
(9.2)
The bending moment about the section centroid in a bent bar state is
(9.3)
Fig. 9.3.
The minus
sign is taken because the bar is bent with a concave downward while
the deflection
is
positive. If the bar was bent with a concave downward, the deflection
v
would
get the same result. Considering (9.2) the equation (9.3) becomes
Denoting
we get
(9.4)
It is a linear differential equation of the second order. Its general solution as it is well known from mathematics and looks like
(9.5)
Here C and D are constants of integration, to determine these we impose the boundary conditions at the bar ends:
at
2)
at
From the
first condition we get C
=
0.
Hence, the bar is bent by the sine curve
From
the second condition we get
This relation is true for two cases.
The first case D = 0. But if C = 0 and D = 0 as follows from the equation (9.5), the bar deflections are equal to zero, which contradicts the original precondition.
The second
case
sin
This condition is performed when
accepts the following infinite value series:
where n
is any whole number. From this
and since
,
.
Thus we get a countless values number of critical forces corresponding to different shapes of a curved bar.
From a practical point of view the interest is but the minimum value of the critical force when the bar loses its stability.
The first root n=0 does not give the solution to the problem. Under n=1 we receive the minimum value of the critical force:
(9.6)
This is Euler’s formula.
The bar bend along the sine curve with one-half wave corresponds to the critical force, i.e.
The next roots give large values of the critical force, which we will not consider. The bar bends along the sine curve with some half-waves which will be received if the bend along the sine curve with a half-wave is for some reason impossible (for example, because of the intermediate ties existence) corresponds these roots.
It should be kept in mind that the constant D, hence the shape of the bar bend axis properties were indeterminate.
When applying for the research of the elastic buckling not an approximate but the exact differential equation of the bend axis, it is possible to determine both the critical force value and the relation between the compressing force and the bar deflection.