GENERALIZED FORMULA FOR LONGITUDINAL BENDING

I. R. Konyakhin

Usually, in textbooks on strength of materials, Euler’s formula for longitudinal bending is derived from the first case of fixing a column, or else from the second (Fig. 1).

The first case is a rod with one end fixed and the other free; the second case is a rod with hinged ends.

To derive Euler’s formula, the differential equation of the bent axis of the rod is formed, and then from the solution of this equation, as a special case, Euler’s formula is obtained:

\[ P = \frac{\pi^2 EI}{4l^2} \]

or, from the second case,

\[ P = \frac{\pi^2 EI}{l^2}. \]

Euler’s formula can be generalized in the form:

\[ P = \frac{\pi^2 EI}{\alpha l^2}. \]

Here \(\alpha\) is a numerical coefficient characterizing the method of fixing the rod.

Fig. 1

Fig. 2.

Below is given a derivation of the generalized Euler formula, presented in a new form.

Let us imagine a free rod loaded by two equal, oppositely directed forces (Fig. 2). The force \(P\), acting along

the axis of the rod, causes deformation, as shown in Fig. 2. Let us compose the differential equation of the bent axis of the rod:

\[ EIy''=-Py, \]

whence

\[ y''+\frac{P}{EI}y=0. \]

Introduce the notation:

\[ \frac{P}{EI}=k^2, \tag{2} \]

\[ y''+k^2y=0. \tag{3} \]

The solution of this equation will be:

\[ y=C_1\sin kx+C_2\cos kx, \tag{4} \]

where \(C_1\) and \(C_2\) are arbitrary constants of integration, which may be determined from the initial conditions:

for \(x=0,\ y=0\); substituting these values into expression 4, we find

\[ C_2=0, \]

and equation 4 takes the form:

\[ y=C_1\sin kx. \tag{5} \]

The last expression is the equation of a sinusoid whose radius vector is the quantity \(C_1\), and whose scale along the direction of the \(x\)-axis is the quantity \(k\), which shows how many times the wavelength of the sinusoid is smaller than the length of the circle by which this sinusoid is constructed or otherwise formed (Fig. 3).

Fig. 3

The radius vector of this sinusoid is the quantity \(r\); its equation is

\[ y=r\cdot\sin kx, \]

but since in the sine function the argument will be the angle \(\varphi\), it follows that

\[ kx=\varphi. \tag{6} \]

Substituting this expression into the preceding one, we obtain:

\[ y=r\cdot\sin\varphi. \tag{7} \]

This is the equation of the given sinusoid. Earlier a sinusoid was obtained in which, instead of \(r\), there was the quantity \(C_1\); consequently, the arbitrary constant \(C_1\) is the radius vector of the sinusoid along which the rod bends under longitudinal bending. The wavelength of a sinusoid in general may be greater or smaller than the length of the circle \(2\pi r\) that forms it; we shall consider it smaller by \(m\) times, then the length of the sinusoid may be represented as \(m\cdot 2\pi r\).

During the formation of the sinusoid by the motion of point \(A\) (Fig. 3), the path along the \(x\)-axis is in general not equal to the arc \(S\) traversed by the end of the radius vector of the sinusoid \(r\), but is proportional to it:

\[ x = mS, \]

while at the same time

\[ S = r\varphi. \]

Solving these two equations jointly, we find

\[ \varphi = \frac{x}{mr}, \]

whereas earlier it was obtained that

\[ \varphi = kx. \]

From the joint solution of the last two expressions we obtain

\[ mr = \frac{1}{k} \quad \text{or} \quad m \cdot 2\pi r = \frac{2\pi}{k}. \]

Thus the quantity \(k\) is the factor of extension or compression of the sinusoid along the \(x\)-axis. The new form of Euler’s formula will be obtained from the previously found expression (2).

According to the value (6),

\[ k^2 = \frac{\varphi^2}{x^2}, \]

therefore

\[ P = \frac{EI\varphi^2}{x^2}. \tag{8} \]

In view of the fact that the deflection of the rod is small in comparison with its length, the length of the rod may be taken equal to the length of the corresponding part of the wavelength of the sinusoid, i.e.,

\[ x = l, \]

then

\[ P = \frac{EI\varphi^2}{l^2}. \tag{9} \]

In the equation obtained, the quantity \(\varphi\) is the angle formed by the rotation of the radius vector in obtaining a definite segment of the sinusoid: for \(1/4\) of its wave it is \(\frac{\pi}{2}\), for \(1/2\) of its wave \(\varphi = \pi\), etc.

Expression (9) is the generalized Euler formula.

Particular Cases of Euler’s Formula for Various Methods of Fixing Rods

1st case. (Fig. 4).
In this case the length of the rod represents \(1/2\) of a sinusoid wave; consequently, \(\varphi = \frac{\pi}{2}\) (Fig. 3). Substituting this value into formula (9), we obtain

\[ P = \frac{\pi^2 EI}{4l^2}. \tag{10} \]

2nd case. (Fig. 5).

In the second case, the length of the rod is \(1/2\) the wavelength of the sinusoid; therefore, according to Fig. 3, \(\varphi=\pi\); from formula (9) we shall have:

\[ P=\frac{\pi^{2}EI}{l^{2}} . \tag{11} \]

3rd case. (Fig. 6).
In the third case of longitudinal bending, the length of the rod is less than \(3/4\) of the total wavelength of the sinusoid. The length of the rod does not coincide with the axis of the sinusoid and passes from the origin of coordinates tangent to it at a certain point \(A\).

The equation of the sinusoid

\[ y_{\sin}=r\cdot \sin kx . \tag{12} \]

The equation of the tangent to it:

\[ y_{\mathrm{tan}}=ax, \tag{13} \]

where \(a\) is the angular coefficient in the equation of the straight line.

Fig. 4      Fig. 5      Fig. 6

At the point of their tangency

\[ y_{\sin}=y_{\mathrm{tan}}, \]

therefore

\[ r\cdot \sin kx=ax. \tag{14} \]

By the condition of tangency of curves, at the point of their tangency the derivatives are equal:

\[ y'_{\sin}=y'_{\mathrm{tan}}. \]

From expressions 12 and 13 it follows:

\[ y'_{\sin}=r\cdot k\cdot \cos kx, \]

\[ y'_{\mathrm{tan}}=a. \]

Taking the preceding into account, we shall have:

\[ r\cdot k\cdot \cos kx=a. \tag{15} \]

From the simultaneous solution of equations 14 and 15 we obtain:

\[ r \cdot k \cdot \cos kx=\frac{r\cdot \sin kx}{x}, \]

whence

\[ \operatorname{tg} kx=kx. \]

Tangency of the straight line and the curve occurs at \(x=l\), therefore

\[ \operatorname{tg} kl=kl \]

or

\[ \operatorname{tg}\varphi=\varphi. \tag{16} \]

This equation is satisfied by the solution:

\[ \varphi=257^\circ 27'25'' \]

or

\[ \varphi=4{,}49346 \text{ radians} \]

or

\[ \varphi=1{,}43031\pi. \]

Substituting the last value into formula (9), we then have

\[ P=\frac{2{,}04579\pi^2EI}{l^2}, \tag{17} \]

or, approximately:

\[ P=\frac{2\pi^2EI}{l^2}. \tag{18} \]

4th case of fastening of the rod. (Fig. 7).

The length of the rod \(l\) in the fourth case is equal to the full length of a sine-wave. According to Figs. 3 and 7, the angle of formation of a full wave is

\[ \varphi=2\pi, \]

therefore from formula (9) we obtain:

\[ P=\frac{4\pi^2EI}{l^2}. \tag{19} \]

It is also possible to obtain a generalized formula for the critical stress in longitudinal bending

\[ \sigma_{cr}=\frac{P}{F}. \]

Substituting formula (9) here, we obtain

\[ \sigma_{cr}=\frac{EI\varphi^2}{l^2F}, \]

but since

\[ I=i^2F, \]

then

\[ \sigma_{cr}=\frac{Ei^2\varphi^2}{l^2}=\frac{E\varphi^2}{\dfrac{l^2}{i^2}}. \]

Fig. 7

The ratio \(\dfrac{l}{i}=\lambda\) is the slenderness of the bar; consequently,

\[ \sigma_{cr}=\frac{E\varphi^{2}}{\lambda^{2}}. \tag{20} \]

The last expression is a generalized formula for the critical stress in longitudinal bending.

Here \(\varphi\) is the angle of rotation of the radius vector of the sinusoid in forming that part of its wavelength which corresponds to the length of the bar for a given method of its end restraint.

From formula 20 the critical slenderness can be determined as

\[ \lambda_{cr}=\varphi\sqrt{\frac{E}{\sigma_s}}, \tag{21} \]

where \(\sigma_s\) is the yield strength of the material,
\(\lambda_{cr}\) is the critical slenderness, which determines the limit of applicability of Euler’s formula in longitudinal bending.