THEORETICAL STUDY AND COMPARISON OF THE STABILITY OF WEBS IN I-BEAMS OF WELDED AND RIVETED CONSTRUCTION

G. N. Kok.

1. Statement of the Problem and Conclusions

A beam subjected to the action of some system of forces may, at a certain magnitude of the latter, be put out of service either by loss of strength or by loss of stability.

The latter type of deformation is expressed as: buckling of the compressed flange over some portion of its length, twisting of the beam as a whole, and, finally, buckling of the beam web.

The continual improvement of structural material and the corresponding increase in allowable stresses leads, in structures, to the use of elements of continually decreasing thickness.

This circumstance creates favorable conditions for loss of stability by one or another element of the structure.

In order to make full use of the high mechanical qualities of the material, it is necessary, by special measures, to ensure both the local and the overall stability of the beam. Of all the measures used for this purpose, in accordance with the tasks of the present work we shall touch only upon stiffening elements (angles and ribs), which counteract buckling of the web of an I-beam.

At present, built-up beams are made riveted or welded. In the first case the stiffening elements are made of angle iron riveted to the web; in the second, of flat bars welded both to the web and to the flanges.

The distance between stiffening elements is determined from a special chart established by the standards (Unified Standards, 1926 edition). The chart was constructed from data of Prof. S. P. Timoshenko, obtained by him in the theoretical analysis of the stability of a rectangular plate supported along its contour and subjected to the action of shear forces applied to its edges.

During the period corresponding to the construction of the chart, only riveted beams existed in production.

As was indicated above, in riveted beams the stiffening angle is riveted only to the web. As a result of this, it becomes possible for the part of the web enclosed between adjacent stiffening angles and flanges, at the moment of loss of its stability, to obtain along the contour a deflection different from zero. This factor is, in all probability, one of the reasons that led Prof. Timoshenko, in analyzing the stability of the web of a riveted I-beam, to adopt as the calculation scheme a rectangular plate supported along its contour.

At present welded beams have also entered production, the stiffeners of which, as stated above, are welded to the web and to the flanges.

The latter circumstance undoubtedly alters in certain respects the boundary conditions of the web of an I-beam enclosed between the flanges and adjacent stiffening ribs. And although, when spacing the ribs in a welded beam, the existing chart continues to be used, nevertheless many believe that in this case an unnecessarily high safety factor for stability is obtained for the web, since, in their opinion, in terms of boundary conditions the vertical plate is closer to a plate with fixed edges¹, owing to the fact that the stiffening ribs are welded both to the web and to the flanges.

Obviously, by fixity in this case is meant elastic fixity, with a variable coefficient of restraint decreasing toward the middle of the sides, the determination of which does not seem possible without carrying out special experiments.

It is difficult to say to what extent the conception of the work of the plate between stiffening ribs in a welded beam corresponds to the work of a plate fixed along its contour.

Leaving this question open, we set ourselves the following task.

Assuming that a plate with elastic fixity along the contour may be adopted as the design scheme for the web of a welded beam, to determine what influence this has on the magnitude of the critical force as compared with simple support.

Drawing 1.

In view of the extreme mathematical complexity and physical uncertainty of the phenomenon of elastic fixity, we were compelled to confine ourselves to only one particular case of fixity—absolutely rigid.

The values of the critical forces for simple support and for absolute fixity are indicators by means of which one may judge the necessity of sharply distinguishing welded beams from riveted ones when investigating the stability of their webs.

In accordance with this, we proposed to study the question of the stability of a rectangular plate fixed along its contour under the following types of loading (Drawing 1).

By compiling the material available in the literature on the last two problems \((b, c)\), and also using the results obtained by us in solving the first problem \((a)\), the question outlined above can in the main be clarified.

For a rectangular plate supported along the contour and subjected to normal forces varying according to the law of normal stresses in pure bending, according to the data of S. P. Timoshenko, the critical forces are determined by the formula

\[ \sigma_{\mathrm{cr}} = k \cdot \sigma_e = k \frac{\Pi^2 D^{2)}}{b^2 h}, \]

where the values of the coefficient \(k\) may be taken from Table 1.

Table 1.

According to an investigation carried out by the author for a rectangular plate clamped along the contour and subjected to an analogous load, the magnitudes of the critical forces may be taken from Table 2.

Table 2.

If we reduce our data to the form adopted by S. P. Timoshenko, it turns out that, under the action of a normal load, for a rectangular plate fixed along the contour, the magnitude of the critical force increases approximately 2.5 times in comparison with a simply supported plate.

We shall make an analogous comparison for the case of the action of a shear load.

According to S. P. Timoshenko’s data, for a rectangular plate along whose contour shear forces act, the critical value of the latter may be expressed by the formula

\[ \tau_{cr}=k\,\frac{\pi^2D}{b^2h} \]

The values of the coefficient $K$ are taken from Table 3.

Table 3.

With further increase of the ratio $\dfrac{a}{b}$, the quantity $k$ asymptotically approaches a certain constant limit, and for $\dfrac{a}{b}=\infty$, $k=5.35$.

For a clamped plate, under the same type of loading, only one case has been considered in the literature, when $\dfrac{a}{b}=\infty$.

In this case the coefficient \(k = 8.98\) ¹).

Comparing this value of \(k\) with the corresponding value in the case of a simply supported plate, we see that the critical force of an infinitely long plate, when the boundary conditions are changed from support to clamping, increases by a factor of 1.68.

From Table 3 it is directly evident that the critical forces for values

\[ \frac{a}{b}=1.5—3 \]

are comparatively close to the critical force corresponding to an infinitely long plate. There is reason to suppose that a similar regularity will also remain valid for a plate with clamped edges.

If we now recall that, in the arrangement of stiffening ribs, the most characteristic ratios are

\[ \frac{a}{b}=1.5—2.5, \]

then we are entitled to suppose that, for the case of loading a rectangular plate by a tangential load, the critical force for a plate clamped along its contour will be, approximately, twice as large as compared with a simply supported plate, all other conditions being equal.

It should be thought that under a combined load (transverse bending) one may expect a similar increase in the critical forces when passing from a simply supported plate to one clamped along the contour.

Taking into account that in reality in a welded beam there is not absolute but elastic clamping of the part of the web between the flanges and ribs, we are entitled to expect an increase in the critical force for the web of a welded beam considerably less than twofold, as compared with the web of a riveted beam.

Proceeding from all that has been said above, we arrive at the conclusion that, practically, there are no special reasons to draw a distinction between a welded and a riveted beam in studying the question of stability of the vertical sheet.

Below is given the theoretical investigation, carried out by the author, of the problem of stability of a rectangular plate with clamped edges, subjected to the action of forces lying in the middle plane and parallel to the \(X\)-axis.

2. Analysis of the phenomenon of stability under the action of normal stresses.

Let us assume that the intensity of the forces varies according to a linear law determined by the expression

\[ \sigma_x=\sigma\left(1-\frac{b-y}{ab}\right). \]

As positive we take the force coinciding with the direction of displacement of the loaded edges when the plate buckles. We assume the coefficient \(\alpha = 0.5\) (pure bending).

The problem is solved by means of a method based on consideration of the strain energy of the system. The starting point of this method is the expression for the buckled form of equilibrium of the system.

According to the well-known Weierstrass theorem, the equation of any plane curve may be represented in the form of a trigonometric polynomial, and the greater the number of terms taken in this polynomial, the more closely it will approach the equation of the function under consideration.

¹) Serenson, Fundamentals of the Technical Theory of Elasticity.

It is obvious that the equation of any surface can be represented in the form of a double sum,

\[ W=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} f_{mn}\sin\frac{m\pi x}{a}\cdot \sin\frac{n\pi y}{b} \tag{1} \]

It is not difficult to see that the adopted expression for the deflections (1) satisfies the boundary conditions only for a plate simply supported along its contour, and therefore it cannot be useful for our case.

However, on the basis of Ritz’s work, in an approximate solution (with any degree of accuracy) of the problem of unstable equilibrium, it is possible to express the equation of the deflections for a rectilinear bar, at any possible displacement at the moment of loss of stability, through a sum of suitable functions.

The number of terms in the sum depends on the desired degree of accuracy.

\[ W=f_1\varphi_1+f_2\varphi_2+f_3\varphi_3+f_4\varphi_4+ \tag{2} \]

There are no definite rules for selecting the functions \(\varphi_i\), and usually they are chosen on the basis of already known solutions of simpler related cases.

In doing so, only two conditions are required of the functions \(\varphi_i\): 1) each of the functions \(\varphi\) individually must satisfy the boundary conditions of the given problem; 2) the series (2) must be convergent.

Fig. 2

The coefficients \(f_i\), which also represent certain displacements, are determined from the condition of the minimum value of the external forces at the moment when the bar loses its rectilinear form of equilibrium.

To consider the question of the stability of a plate, the equation of the bent middle plane will have to be taken as the product of two series determined by expression (2).

The accuracy of the obtained solution can be established by adding an extra term to the right-hand side of expression (2) and considering how much this affects the result.

To write the equation of the deflections of the plate under consideration, we shall use the considerations set forth above.

Proceeding from the boundary conditions, the condition of convergence of the series, and the expected form of the curved middle plane (symmetry in the direction of the axis ...

X-axes and asymmetry in the direction of the \(Y\)-axes), we take the equation of deflections of the plate under consideration in the following form:

\[ W=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} f_{mn}\sin^{2}\frac{m\pi x}{a}\cdot\frac{y}{b}\cdot\sin^{2}\frac{n\pi y}{b} \tag{3} \]

This equation will correspond to a whole series of curved equilibrium forms, depending on how many terms are taken in each of the sums.

To each of these curved equilibrium forms there will correspond its own critical force.

We are interested in the smallest of these forces, i.e., the truly critical one, corresponding to the first possible curved equilibrium form of the plate.

It is quite obvious that this force will correspond to such a buckling of the middle plane in the direction of the \(X\)-axis for which the curve forming the half-wave in the same direction will have a smooth variation of the first derivative. And this can occur provided that the equation of the curved middle plane in the direction of the \(X\)-axis is expressed not by a sum of terms \(\sum_{m=1}^{\infty} f_m\sin^{2}\frac{m\pi x}{a}\), but only by one term

\[ f_m\sin^{2}\frac{m\pi x}{a}, \]

where \(m\) is the number of half-waves into which the plate is subdivided when buckling along the \(X\)-axis.

Each half-wave may be considered as an independent plate; therefore, in what follows, we may confine ourselves to studying the buckling of the plate over one half-wave, i.e., take \(m=1\).

The final expression for the curved middle plane is taken in the form

\[ w=f_1\sin^{2}\frac{\pi x}{a}\sum_{n=1}^{\infty} f_n\,\frac{y}{b}\sin^{2}\frac{n\pi y}{b} \tag{4} \]

Let us verify that the chosen deflection function satisfies the boundary conditions.

\[ \frac{dw}{dx} = \frac{\pi}{a\cdot b}\,f_1\cdot\sin 2\frac{\pi x}{a} \sum_{n=1}^{\infty} f_n\cdot y\cdot\sin^{2}\frac{n\pi y}{b} \]

\[ \frac{dw}{dy} = f_1\cdot\sin^{2}\frac{\pi x}{a} \sum_{n=1}^{\infty} f_n\cdot\frac{1}{b} \left( \sin^{2}\frac{n\pi y}{b} + \frac{n\pi}{b}\,y\cdot\sin 2\frac{n\pi y}{b} \right) \]

\[ \begin{array}{ll} \text{for } x=0\left\{\begin{array}{l} w=0\\[2pt] \dfrac{dw}{dx}=0 \end{array}\right. & \text{for } y=0\left\{\begin{array}{l} w=0\\[2pt] \dfrac{dw}{dy}=0 \end{array}\right. \\[14pt] \phantom{\text{for }}x=a\left\{\begin{array}{l} \\[-10pt] \end{array}\right. & \phantom{\text{for }}y=b\left\{\begin{array}{l} \\[-10pt] \end{array}\right. \end{array} \]

We shall briefly set forth the essence of the method adopted by us for solving the problem posed.

If, for every possible deviation of the plate from the plane form of equilibrium, the increment of the potential energy of deformation is greater than the work of the external forces which they perform as a result of displacement of the edges during curvature of the plate, then the plane form is stable.

In the opposite case it will be unstable.

Obviously, at the moment when the potential energy of deformation is equal to the work of the external forces, the plate will be in an indifferent form of equilibrium, i.e., it will stand on the verge of losing stability. The magnitude of the external forces corresponding to this moment will represent their critical value.

If we denote by \(V\) the potential energy of bending corresponding to a very small deviation of the plate from the plane form of equilibrium, and by \(T\) the work of the external forces on these deviations, then the critical value of the compressive forces can be found from the condition:

\[ V = T \quad \text{or} \quad T - V = 0 \tag{5} \]

Both the potential energy of bending and the work of the external forces are expressed, by means of the corresponding formulas given below, through the deflection function \((w)\); thus \(V\) and \(T\) are functions of the unknown coefficients \(f_i\).

In general form this dependence may be represented as follows:

\[ T = \sigma_{cr} F(f_1, f_2, f_3 \ldots \ldots) \]

\[ U = F_1(f_1, f_2, f_3 \ldots \ldots) \]

Therefore equation (5) takes the form

\[ \sigma_{cr} F(f_1, f_2, f_3 \ldots \ldots) - F_1(f_1, f_2, f_3 \ldots \ldots) = 0, \]

whence

\[ \sigma_{cr} = \frac{F_1(f_1, f_2, f_3 \ldots \ldots)} {F(f_1, f_2, f_3 \ldots \ldots)} \tag{6} \]

In accordance with the problem posed, the coefficients \(f_i\) must be chosen so that \(\sigma_{cr}\) is a minimum. The equations for determining the coefficients will have the form

\[ \frac{d\sigma_{cr}}{df} = \frac{dF_1}{df}\cdot F - \frac{dF}{df} F_1 =0 \tag{7} \]

and there will be as many of them as there are unknown coefficients.

The very same equations serving to determine the coefficients can be obtained if, as the basis of the consideration, one takes not equation (5), but the expression for the change in the potential energy of the entire system under study, \(V_0\) (i.e., the elastic system plus the external forces), which may be written as \(V_0 = T - V\), or

\[ V_0 = \sigma_{cr} F(f_1, f_2, f_3 \ldots \ldots) - F_1(f_1, f_2, f_3 \ldots \ldots) \tag{8} \]

The unknown coefficients \(f_i\) can be determined from the condition that the change in the potential energy of the entire system be a minimum.

\[ \frac{dV_0}{df} = \frac{d(T - V)}{df} =0 \tag{9} \]

This is the path we shall follow in the further investigation.

The condition \(\dfrac{dV_0}{df}=0\), with \(n\) unknown coefficients, leads to a system of \(n\) linear homogeneous equations, in which, in addition to the unknown coefficients \(f_i\), the quantity \(\sigma_{cr}\) will also be unknown.

In order for the coefficients \(f_i\) to be different from zero—and only this case can be of interest to us—it is necessary that the determinant of this system of equations be equal to zero.

From this condition the value of \(\sigma_p^k\) is also determined.

We now proceed directly to the determination of \(\sigma_{cr}\) for our case.

The general expression for the potential energy of a bent plate may be represented in the following form:

\[ V=\frac{D}{2}\int_{0}^{a}\int_{0}^{b} \left\{ \left( \frac{d^{2}W}{dx^{2}}+\frac{d^{2}W}{dy^{2}} \right)^{2} - 2(1-\mu) \left[ \frac{d^{2}W}{dx^{2}}\cdot\frac{d^{2}W}{dy^{2}} - \left( \frac{d^{2}W}{dx\cdot dy} \right)^{2} \right] \right\}\,dx\,dy \tag{10} \]

where

\[ D=\frac{Eh^{3}}{12(1-\mu^{2})} \]

is the cylindrical stiffness of the plate,

\(w\) — deflection function,
\(\mu\) — Poisson’s ratio,
\(h\) — thickness of the plate.

The general expression for the work of external forces applied to the transverse edges and directed parallel to the \(X\)-axis may be represented in the following form:

\[ T=-\frac{1}{2}\int_{0}^{a}dy\int_{0}^{b}p \left( \frac{dW}{dx} \right)^{2}\,dx, \tag{11} \]

where \(p\) is the intensity of the external force; for our case we set

\[ p=\sigma_x=\sigma\left(1-\frac{b-y}{\alpha b}\right) \]

\(w\) — deflection function.

We find the value of \(V\) in accordance with the adopted equation of deflections. The values of the second partial derivatives are:

\[ \frac{\partial^{2}w}{\partial x^{2}} = -\frac{2\pi^{2}}{a^{2}b} \sum_{n=1}^{\infty} f_{1n}\cos 2\frac{\pi x}{a}\cdot y\sin^{2}\frac{n\pi y}{b} \]

\[ \frac{\partial^{2}w}{\partial y^{2}} = 2\frac{\pi}{b^{2}} \sum_{n=1}^{\infty} f_{1n}n\cdot \sin^{2}\frac{\pi x}{a} \left( \sin 2\frac{n\pi y}{b} + \frac{n\pi}{b}\cdot y\cos 2\frac{n\pi y}{b} \right) \]

\[ \frac{\partial^{2}w}{\partial x\cdot\partial y} = \frac{\pi}{ab} \sum_{n=1}^{\infty} f_{1n}\cdot \sin 2\frac{\pi x}{a} \left( \sin^{2}\frac{n\pi y}{b} + y\frac{n\pi}{b}\cdot\sin 2\frac{n\pi y}{b} \right) \]

The first term of the integrand expression for \(V\) is

\[ \left( \frac{\partial^{2}w}{\partial x^{2}} + \frac{\partial^{2}w}{\partial y^{2}} \right)^{2} = \left[ \frac{2\pi^{2}}{a^{2}b} \sum_{n=1}^{\infty} f_{1n}\cos 2\frac{\pi x}{a}\cdot y\sin^{2}\frac{n\pi y}{b} + \right. \]

\(^1\) Timoshenko, Course of the Theory of Elasticity, Part II.

\[ \begin{aligned} &+\frac{2\pi}{b^{2}}\sum_{n=1}^{\infty} f_{1n} n\sin^{2}\frac{\pi x}{a} \left(\sin 2\frac{n\pi y}{b}+y\frac{n\pi}{b}\cos 2\frac{n\pi y}{b}\right)\Bigg]^{2} = \\[4pt] &=\frac{4\pi^{4}}{a^{4}b^{2}}\sum_{n=1}^{\infty} f_{1n}^{2}\cos^{2}2\frac{\pi x}{a}\cdot y^{2}\sin^{4}\frac{n\pi y}{b}+ \\[4pt] &\quad+\frac{4\pi^{2}}{b^{4}}\sum_{n=1}^{\infty} f_{1n}^{2}\cdot n^{2}\sin^{4}\frac{\pi x}{a} \left(\sin 2\frac{n\pi y}{b}+y\frac{n\pi}{b}\cos 2\frac{n\pi y}{b}\right)^{2}+ \\[4pt] &\quad+2\frac{4\pi^{4}}{a^{4}b^{2}}\sum_{n=i=1}^{\infty} f_{1n}f_{1i}\cos^{2}2\frac{\pi x}{a}\,y^{2}\cdot \sin^{2}\frac{n\pi y}{b}\cdot\sin^{2}\frac{i\pi y}{b}+ \\[4pt] &\quad+2\frac{4\pi^{2}}{b^{4}}\sum_{n\ne i=1}^{\infty} f_{1n}f_{1i}\cdot ni\,\sin^{4}\frac{\pi x}{a} \left(\sin 2\frac{n\pi y}{b}+y\frac{n\pi}{b}\cos 2\frac{n\pi y}{b}\right)\cdot \\[4pt] &\qquad\qquad\cdot \left(\sin 2\frac{i\pi y}{b}+y\frac{i\pi}{b}\cos 2\frac{i\pi y}{b}\right)+ \\[4pt] &\quad+2\frac{4\pi^{3}}{a^{2}b^{3}}\sum_{n=1}^{\infty} f_{1n}^{2}\cdot n\cos 2\frac{\pi x}{a}\cdot\sin^{2}\frac{\pi x}{a}\cdot y\cdot \sin^{2}\frac{n\pi y}{b} \left(\sin 2\frac{n\pi y}{b}+\right. \\[4pt] &\left.\qquad\qquad +y\frac{n\pi}{b}\cos 2\frac{n\pi y}{b}\right) +2\frac{4\pi^{3}}{a^{2}b^{3}}\sum_{n\ne i=1}^{\infty} f_{1n}f_{1i}\cdot i\cos 2\frac{\pi x}{a}\cdot\sin^{2}\frac{\pi x}{a}\cdot y\sin^{2}\frac{n\pi y}{b} \left(\sin 2\frac{i\pi y}{b}+\right. \\[4pt] &\left.\qquad\qquad +y\frac{i\pi}{b}\cos 2\frac{i\pi y}{b}\right). \end{aligned} \]

Passing to double integration of the expressions standing under the summation sign \(\Sigma\), we shall have:

\[ \int_{0}^{a}\int_{0}^{b} f_{1n}^{2}\cos^{2}2\frac{\pi x}{a}\cdot y^{2}\sin^{4}\frac{n\pi y}{b}\,dx\,dy = f_{1n}^{2}\frac{ab^{3}}{16} \left(1-\frac{15}{8n^{2}\pi^{2}}\right) \]

\[ \int_{0}^{a}\int_{0}^{b} f_{1n}^{2}\cdot n^{2}\sin^{4}\frac{\pi x}{a} \left(\sin 2\frac{n\pi y}{b}+y\frac{n\pi}{b}\cos 2\frac{n\pi y}{b}\right)^{2} \,dx\,dy = \]

\[ = f_{1n}^{2}\cdot n^{2}\frac{3ab}{16} \left(\frac{5}{8}+\frac{n^{2}\pi^{2}}{3}\right) \]

\[ \int_{0}^{a}\int_{0}^{b} f_{1n}f_{1i}\cos^{2}2\frac{\pi x}{a}\cdot y^{2}\cdot \sin^{2}\frac{n\pi y}{b}\cdot\sin^{2}\frac{i\pi y}{b}\,dx\,dy = \]

\[ f_{1n}f_{1i}\frac{ab^{3}}{8} \left[ \frac{1}{3} -\frac{n^{2}+i^{2}}{2i^{2}n^{2}\pi^{2}} +\frac{n^{2}+i^{2}}{2\left(n^{2}-i^{2}\right)^{2}\pi^{2}} \right] \]

\[ \int_0^a \int_0^b f_{1n} f_{1i}\, n i \,\sin^4 \frac{\pi x}{a} \left( \sin 2 \frac{n\pi y}{b} + y \frac{n\pi}{b} \cos 2 \frac{n\pi y}{b} \right) \cdot \left( \sin 2 \frac{i\pi y}{b} + y \frac{i\pi}{b} \cos 2 \frac{i\pi y}{b} \right) \, dx\,dy = f_{1n} f_{1i}\,\frac{3ab}{16}\, \frac{n^2 i^2 (n^2+i^2)}{(n^2-i^2)^2} \]

\[ \int_0^a \int_0^b f_{1n}^2 n \cos 2 \frac{\pi x}{a}\cdot \sin^2 \frac{\pi x}{a} \cdot y \sin^2 \frac{n\pi y}{b} \left( \sin 2 \frac{n\pi y}{b} + y \frac{n\pi}{b} \cos 2 \frac{n\pi y}{b} \right) \, dx\,dy = - f_{1n}^2 n \frac{ab^2}{16} \left( \frac{1}{8n\pi} - \frac{n\pi}{3} \right). \]

\[ \int_0^a \int_0^b f_{1n} f_{1i}\, i \cos 2 \frac{\pi x}{a}\cdot \sin^2 \frac{\pi x}{a} \cdot y \sin^2 \frac{n\pi y}{b} \left( \sin 2 \frac{i\pi y}{b} + y \frac{i\pi}{b} \cos 2 \frac{i\pi y}{b} \right) \, dx\,dy = f_{1n} f_{1i}\,\frac{ab^2}{8\pi}\, \frac{n^2 i^2}{(i^2-n^2)^2} \]

The second term of the integrand \(V\)

\[ \frac{d^2 w}{dx^2}\cdot \frac{d^2 w}{dy^2} = \left[ \frac{2\pi^2}{a^2 b} \sum_{n=1}^{\infty} f_{1n}\cos 2 \frac{\pi x}{a}\cdot y \sin^2 \frac{n\pi y}{b} \right] \cdot \left[ \frac{2\pi}{b^2} \sum_{n=1}^{\infty} f_{in}\cdot \right. \]

\[ \left. \cdot n \sin^2 \frac{\pi x}{a} \left( \sin 2 \frac{n\pi y}{b} + y \frac{n\pi}{b} \cos 2 \frac{n\pi y}{b} \right) \right] = \frac{4\pi^3}{a^2 b^3} \sum_{n=1}^{\infty} f_{1n}^2 \]

\[ \cdot n \cos 2 \frac{\pi x}{a}\cdot \sin^2 \frac{\pi x}{a} \cdot y \sin^2 \frac{n\pi y}{b} \left( \sin 2 \frac{n\pi y}{b} + y \frac{n\pi}{b} \cos 2 \frac{n\pi y}{b} \right) + \]

\[ + 2 \frac{4\pi^3}{a^2 b^3} \sum_{\substack{n=1\\ i=1}}^{\infty} f_{1n} f_{1i}\, i \cos 2 \frac{\pi x}{a}\cdot \sin^2 \frac{\pi x}{a} \cdot y \sin^2 \frac{n\pi y}{b} \left( \sin 2 \frac{i\pi y}{b} + y \frac{i\pi}{b} \cos 2 \frac{i\pi y}{b} \right) \]

We carry out double integration of the expressions standing under the sign \(\Sigma\)

\[ \int_0^a \int_0^b f_{1n}^2 \cdot n \cos 2 \frac{\pi x}{a}\cdot \sin^2 \frac{\pi x}{a} \cdot y \sin^2 \frac{n\pi y}{b} \left( \sin 2 \frac{n\pi y}{b} + y \frac{n\pi}{b} \cos 2 \frac{n\pi y}{b} \right) \, dx\,dy = - f_{1n}^2 \cdot n \frac{ab^2}{16} \left( \frac{1}{8n\pi} - \frac{n\pi}{3} \right) \]

\[ \int_0^a \int_0^b f_{1n} f_{1i}\, i \cos 2 \frac{\pi x}{a}\cdot \sin^2 \frac{\pi x}{a} \cdot y \sin^2 \frac{n\pi y}{b} \left( \sin 2 \frac{i\pi y}{b} + y \frac{i\pi}{b} \cos 2 \frac{i\pi y}{b} \right) \, dx\,dy = f_{1n} f_{1i}\,\frac{ab^2}{8\pi}\, \frac{n^2 i^2}{(i^2-n^2)^2} \]

The third term of the integrand of \(V\)

\[ \left(\frac{d^{2}w}{dx\,dy}\right)^{2} = \left[ \frac{\pi}{ab}\sum_{n=1}^{\infty} f_{1n}\sin 2\frac{\pi x}{a} \left( \sin^{2}\frac{n\pi y}{b} + y\frac{n\pi}{b}\cdot \sin 2\frac{n\pi y}{b} \right) \right]^{2} \]

\[ = \frac{\pi^{2}}{a^{2}b^{2}} \sum_{n=1}^{\infty} f^{2}_{1n}\sin^{2}2\frac{\pi x}{a} \left( \sin^{2}\frac{n\pi y}{b} + y\frac{n\pi}{b}\cdot \sin 2\frac{n\pi y}{b} \right)^{2} + 2\frac{\pi^{2}}{a^{2}b^{2}} \sum_{n=1}^{\infty} f_{1n}f_{1i}\sin^{2}2\frac{\pi x}{a} \left( \sin^{2}\frac{n\pi y}{b} + y\frac{n\pi}{b}\sin 2\frac{n\pi y}{b} \right) \left( \sin^{2}\frac{i\pi y}{b} + y\frac{i\pi}{b}\sin 2\frac{i\pi y}{b} \right) \]

We carry out the double integration of the expressions standing under the sign \(\Sigma\).

\[ \int_{0}^{a}\int_{0}^{b} f^{2}_{1n}\sin^{2}2\frac{\pi x}{a} \left( \sin^{2}\frac{n\pi y}{b} + y\frac{n\pi}{b}\cdot \sin 2\frac{n\pi y}{b} \right)^{2} dx\,dy = f^{2}_{1n}\frac{ab}{4} \left( \frac{n^{2}\pi^{2}}{3}-\frac{1}{8} \right) \]

\[ \int_{0}^{a}\int_{0}^{b} f_{1n}f_{1i}\sin^{2}2\frac{\pi x}{a} \left( \sin^{2}\frac{n\pi y}{b} + y\frac{n\pi}{b}\cdot \sin 2\frac{n\pi y}{b} \right) \left( \sin^{2}\frac{i\pi y}{b} + y\frac{i\pi}{b}\sin 2\frac{i\pi y}{b} \right) dx\,dy = f_{1n}f_{1i}\frac{ab}{2}\, \frac{n^{2}i^{2}}{(n^{2}-i^{2})^{2}} \]

The expression for the potential energy of bending takes the form

\[ V= \frac{Eh^{3}}{24(1-\mu^{2})} \left\{ \left[ \frac{4\pi^{4}}{a^{4}b^{2}}\cdot\frac{ab^{3}}{16} \sum_{n=1}^{\infty} f^{2}_{1n} \left( 1-\frac{15}{8n^{2}\pi^{2}} \right) \right] + \frac{4\pi^{2}}{b^{4}}\cdot\frac{3ab}{16} \sum_{n=1}^{\infty} f^{2}_{1n}n^{2} \left( \frac{5}{8}+\frac{n^{2}\pi^{2}}{3} \right) + \right. \]

\[ \left. + 2\frac{4\pi^{4}}{a^{4}b^{2}}\cdot\frac{ab^{3}}{8} \sum_{n\ne i} f_{1n}f_{1i} \left( \frac{1}{3} - \frac{n^{2}+i^{2}}{2i^{2}n^{2}\pi^{2}} + \frac{n^{2}+i^{2}}{2(n^{2}-i^{2})^{2}\pi^{2}} \right) + \right. \]

\[ \left. + 2\frac{4\pi^{2}}{b^{4}}\cdot\frac{3ab}{16} \sum_{n\ne i} f_{1n}f_{1i} \frac{n^{2}i^{2}(n^{2}+i^{2})}{(n^{2}-i^{2})^{2}} - 2\frac{4\pi^{3}}{a^{2}b^{3}}\cdot\frac{ab^{2}}{16} \sum_{n=1}^{\infty} f^{2}_{1n}n \left( \frac{1}{8n\pi}-\frac{n\pi}{3} \right) + \right. \]

\[ \left. + 2\frac{4\pi^{3}}{a^{2}b^{3}}\cdot\frac{ab^{2}}{8\pi} \sum_{n\ne i} f_{1n}f_{1i} \frac{n^{2}i^{2}}{(i^{2}-n^{2})^{2}} \right] - \]

$$ -2(1-\mu)\left[-\frac{4\pi^{3}}{a^{2}b^{3}}\frac{ab^{2}}{16}\sum_{n=1}^{\infty} f_{1n}^{2} n\left(\frac{1}{8n\pi}-\frac{n\pi}{3}\right)+\right. $$

$$ +2\frac{4\pi^{3}}{a^{2}b^{3}}\cdot\frac{ab^{2}}{8\pi}\cdot \sum_{n\ne i}^{\infty} f_{1n}f_{1i}\frac{n^{2}i^{2}}{(i^{2}-n^{2})^{2}}- $$

$$ \left. -\frac{\pi^{2}}{a^{2}b^{2}}\frac{ab}{4}\sum_{n=1}^{\infty} f_{1n}^{2} \left(\frac{n^{2}\pi^{2}}{3}-\frac{1}{8}\right) -2\frac{\pi^{2}}{a^{2}b^{2}}\frac{ab}{2}\sum_{n=i}^{\infty} f_{1n}f_{1i}\frac{n^{2}i^{2}}{(n^{2}-i^{2})^{2}} \right\} $$

As is not difficult to see, the expression standing in the second square brackets becomes zero. After the corresponding transformation, the final value of \(V\) takes the form

$$ V=\frac{Eh^{3}\pi^{2}}{24(1-\mu^{2})ab}\left\{ \left[ \frac{\pi^{2}}{4}-\frac{b^{2}}{a^{2}}\sum_{n=1}^{\infty} \left(1-\frac{15}{8n^{2}\pi^{2}}\right)+ \right.\right. $$

$$ \left. +\frac{3}{4}\frac{a^{2}}{b^{2}}\sum_{n=1}^{\infty} n^{2}\left(\frac{5}{8}+\frac{n^{2}\pi^{2}}{3}\right) +\frac{1}{2}\sum_{n=1}^{\infty} \left(\frac{n^{2}\pi^{2}}{3}-\frac{1}{8}\right) \right] f_{1n}^{2}+ $$

$$ +\left[ \frac{\pi^{2}b^{2}}{a^{2}}\sum_{n\ne i}^{\infty} \left( \frac{1}{3}-\frac{n^{2}+i^{2}}{2i^{2}n^{2}\pi^{2}} +\frac{n^{2}+i^{2}}{2(n^{2}-i^{2})^{2}\pi^{2}} \right) +\frac{3}{2}\frac{a^{2}}{b^{2}}\sum_{n\ne i}^{\infty} \frac{n^{2}i^{2}(n^{2}+i^{2})}{(n^{2}-i^{2})^{2}}+ \right. $$

$$ \left. +\sum_{n\ne i}^{\infty}\frac{n^{2}i^{2}}{(i^{2}-n^{2})^{2}} \right] f_{1n}f_{1i} \right\} \tag{12} $$

Let us determine the work of the external forces

$$ T=\frac{1}{2}\int_{0}^{b}dy\int_{0}^{a} \sigma\left(1-\frac{b-y}{ab}\right) \left(\frac{dw}{dx}\right)^{2}dx $$

$$ T=\frac{\sigma}{2}\left[ \left(1-\frac{1}{a}\right)\int_{0}^{a}\int_{0}^{b} \left(\frac{dw}{dx}\right)^{2}dx\,dy +\frac{1}{ab}\int_{0}^{a}\int_{0}^{b} y\left(\frac{dw}{dx}\right)^{2}dx\,dy \right] \tag{13} $$

For the case under consideration

$$ \left(\frac{dw}{dx}\right)^{2} =\left( \frac{\pi}{ab}\sum_{n=1}^{\infty} f_{n1}\sin 2\frac{\pi x}{a}\cdot y\sin^{2}\frac{n\pi y}{b} \right)^{2}= $$

$$ =\frac{\pi^{2}}{a^{2}b^{2}}\sum_{n=1}^{\infty} f_{1n}^{2}\sin^{2}2\frac{\pi x}{a}\,y^{2}\sin^{4}\frac{n\pi y}{b}+ $$

$$ +2\frac{\pi^{2}}{a^{2}b^{2}}\sum_{n\ne i}^{\infty} f_{1n}f_{1i}\sin^{2}2\frac{\pi x}{a}\,y^{2} \sin^{2}\frac{n\pi y}{b}\cdot\sin^{2}\frac{i\pi y}{b} $$

\[ \int_{0}^{a}\int_{0}^{b} f_{1n}^{2}\sin^{2}2\,\frac{\pi x}{a}\,y^{2}\sin^{4}\frac{n\pi y}{b}\,dx\,dy = f_{1n}^{2}\frac{ab^{3}}{16}\left(1-\frac{15}{8n^{2}\pi^{2}}\right) \]

\[ \int_{0}^{a}\int_{0}^{b} f_{1n}f_{1i}\sin^{2}2\,\frac{\pi x}{a}\,y^{2}\sin^{2}\frac{n\pi y}{b}\cdot\sin^{2}\frac{i\pi y}{b}\,dx\,dy = f_{1n}f_{1i}\frac{ab^{3}}{8}\left( \frac{1}{3} -\frac{n^{2}+i^{2}}{2i^{2}n^{2}\pi^{2}} +\frac{n^{2}+i^{2}}{2(n^{2}-i^{2})^{2}\pi^{2}} \right) \]

\[ y\left(\frac{dw}{dx}\right)^{2} = \frac{\pi^{2}}{a^{2}b^{2}}\sum_{n=1}^{\infty} f_{1n}^{2}\sin^{2}2\,\frac{\pi x}{a}\,y^{3}\sin^{4}\frac{n\pi y}{b} + \]

\[ +2\,\frac{\pi^{2}}{a^{2}b^{2}}\sum_{n\ne i}^{\infty} f_{1n}f_{1i}\sin^{2}2\,\frac{\pi x}{a}\,y^{3}\sin^{2}\frac{n\pi y}{b}\cdot\sin^{2}\frac{i\pi y}{b} \]

\[ \int_{0}^{a}\int_{0}^{b} f_{1n}^{2}\sin^{2}2\,\frac{\pi x}{a}\cdot y^{3}\sin^{4}\frac{n\pi y}{b}\,dx\,dy = f_{1n}^{2}\frac{3ab^{4}}{64} \left(1-\frac{15}{4n^{2}\pi^{2}}\right) \]

\[ \int_{0}^{a}\int_{0}^{b} f_{1n}f_{1i}\sin^{2}2\,\frac{\pi x}{a}\,y^{3}\sin^{2}\frac{n\pi y}{b}\cdot\sin^{2}\frac{i\pi y}{b}\,dx\,dy = \]

\[ = f_{1n}f_{1i}\frac{ab^{4}}{32} \left( 1-\frac{3(n^{2}+i^{2})}{i^{2}n^{2}\pi^{2}} +\frac{3(n^{2}+i^{2})}{(n^{2}-i^{2})^{2}\pi^{2}} \right) \]

Substituting the found values of the integrals into expression 13, after transformation we obtain the final expression for \(T\) in the following form

\[ T=\frac{\sigma\pi^{2}b}{8a} \left\{ \left[ \left(1-\frac{1}{\alpha}\right)\frac{1}{4} \sum_{n=1}^{\infty}\left(1-\frac{15}{8n^{2}\pi^{2}}\right) + \frac{1}{\alpha}\frac{3}{16} \sum_{n=1}^{\infty}\left(1-\frac{15}{4n^{2}\pi^{2}}\right) \right]f_{1n}^{2} + \right. \]

\[ + \left[ \left(1-\frac{1}{\alpha}\right) \sum_{n\ne i}^{\infty} \left( \frac{1}{3} -\frac{n^{2}+i^{2}}{2i^{2}n^{2}\pi^{2}} +\frac{n^{2}+i^{2}}{2(n^{2}-i^{2})^{2}\pi^{2}} \right) + \right. \]

\[ \left. \left. +\frac{1}{4\alpha} \sum_{n\ne i}^{\infty} \left( 1-\frac{3(n^{2}+i^{2})}{i^{2}n^{2}\pi^{2}} +\frac{3(n^{2}+i^{2})}{(n^{2}-i^{2})^{2}\pi^{2}} \right) \right]f_{1n}f_{1i} \right\} \tag{14} \]

We find the change in the potential energy of the entire system, for which we substitute expressions (12) and (14) into (8)

\[ V_{0}= \frac{\sigma\pi^{2}b}{8a} \left\{ \left[ \left(1-\frac{1}{\alpha}\right)\frac{1}{4} \sum_{n=1}^{\infty}\left(1-\frac{15}{8n^{2}\pi^{2}}\right) + \frac{1}{\alpha}\frac{3}{16} \sum_{n=1}^{\infty}\left(1-\frac{15}{4n^{2}\pi^{2}}\right) \right]f_{1n}^{2} + \right. \]

\[ +\left[\left(1-\frac{1}{\alpha}\right)\sum_{n\ne 1}^{\infty}\left(\frac{1}{3}-\frac{n^{2}+i^{2}}{2i^{2}n^{2}\pi^{2}}+\frac{n^{2}+i^{2}}{2(n^{2}-i^{2})^{2}\pi^{2}}\right)+ \right. \]
\[ \left. +\frac{1}{4\alpha}\sum_{n\ne i}^{\infty}\left(1-\frac{3(n^{2}+i_{2})}{i^{2}n^{2}\pi^{2}}+\frac{3(n^{2}+i^{2})}{(n^{2}-i^{2})^{2}\pi^{2}}\right)\right] f_{1n}f_{1i}\right\} \]
\[ -\frac{Eh^{3}\pi^{2}}{24(1-\mu^{2})ab}\left\{ \frac{\pi^{2}}{4}\frac{b^{2}}{a^{2}}\sum_{n\ne i}^{\infty}\left(1-\frac{15}{8n^{2}\pi^{2}}\right) +\frac{3}{4}\frac{a^{2}}{b^{2}}\sum_{n\ne i}^{\infty}h^{2}\left(\frac{5}{8}+\frac{n^{2}\pi^{2}}{3}\right)+ \right. \]
\[ \left. +\frac{1}{2}\sum_{n\ne i}^{\infty}\left(\frac{n^{2}\pi^{2}}{3}-\frac{1}{8}\right)\right]f_{1n}^{2}+ \]
\[ +\left[\frac{\pi^{2}b^{3}}{a^{2}}\sum_{n\ne i}^{\infty}\left(\frac{1}{3}-\frac{n^{2}+i^{2}}{2i^{2}n^{2}\pi^{2}}+\frac{n^{2}+i^{2}}{2(n^{2}-i^{2})^{2}\pi^{2}}\right) +\frac{3}{2}\frac{a^{2}}{b^{2}}\sum_{n\ne i}^{\infty}\frac{n^{2}i^{2}(n^{2}+i^{2})}{(n^{2}-i^{2})^{2}}+ \right. \]
\[ \left. +2\sum_{n\ne i}^{\infty}\frac{n^{2}i^{2}}{(i^{2}-n^{2})^{2}}\right]f_{1n}f_{1i}\right\} \]

In order, among all possible forms of the curved equilibrium, to choose that to which the smallest value of \(\sigma_{cr}\) corresponds, it is necessary to select the coefficients \(f_i\) from the condition of a minimum change in the potential energy of the entire system.

As was indicated above, this condition leads to the solution of the corresponding determinant, from which we find the value of \(\sigma_{cr}\) corresponding to the chosen function \(W\).

By successive approximations, i.e., by gradually increasing the number of terms in the expression for \(W\), the minimum value of \(\sigma_{cr}\) can be found; this is the final solution of the problem posed.

By considering a series of approximations, we established that, in order to obtain comparatively accurate results, it is sufficient to consider the third approximation. For this case the values of \(n\) are taken equal to \(1, 2, 3\).

After a series of transformations, taking into account that \(\alpha=0.5\) and
\[ F=\frac{Eh^{3}}{3(1-\mu^{2})ab}, \]
we reduce the equation for the change in the potential energy of the system to the following form:
\[ \begin{aligned} V_{0}={}& \left[-F\left(2\frac{b^{2}}{a^{2}}+2.94\frac{a^{2}}{b^{2}}+1.58\right)+0.034\frac{\sigma b}{a}\right]f_{11}^{2} +\left[-F\left(2.34\frac{b^{2}}{a^{2}}+\right.\right.\\ &\left.\left.+41.475\frac{a^{2}}{b^{2}}+6.5\right)+0.102\frac{\sigma b}{a}\right]f_{12}^{2} +\left[-F\left(2.42\frac{b^{2}}{a^{2}}+204\frac{a^{2}}{b^{2}}+\right.\right.\\ &\left.\left.+14.78\right)+0.115\frac{\sigma b}{a}\right]f_{13}^{2} +\left[-F\left(2.94\frac{b^{2}}{a^{2}}+3.33\frac{a^{2}}{b^{2}}+0.89\right)+\right.\\ &\left.+0.096\frac{\sigma b}{a}\right]f_{11}f_{12} +\left[-F\left(2.8\frac{b^{2}}{a^{2}}+2.01\frac{a^{2}}{b^{2}}+0.28\right)+0.07\frac{\sigma b}{a}\right]f_{11}f_{13}+\\ &+\left[-F\left(3.36\frac{b^{2}}{a^{2}}+2.8\frac{a^{2}}{b^{2}}+2.88\right)+0.16\frac{\sigma b}{a}\right]f_{12}f_{13}. \end{aligned} \tag{16} \]

The conditions for determining the unknown coefficients \(f_{11}, f_{12}, f_{13}\) are written as follows:

\[ \frac{dV_o}{df_{11}} = 2\left[-F\left(2\frac{b^2}{a^2}+2.94\frac{a^2}{b^2}+1.58\right)+0.034\frac{\sigma b}{a}\right]f_{11} + \left[-F\left(2.93\frac{b^2}{a^2}+3.33\frac{a^2}{b^2}+0.89\right) +0.096\frac{\sigma b}{a}\right]f_{12} + \left[-F\left(2.8\frac{b^2}{a^2}+2.01\frac{a^2}{b^2}+0.28\right) +0.07\frac{\sigma b}{a}\right]f_{13}=0 \tag{17} \]

\[ \frac{dV_o}{df_{12}} = 2\left[-F\left(2.34\frac{b^2}{a^2}+41.475\frac{a^2}{b^2}+6.5\right)+0.102\frac{\sigma b}{a}\right]f_{12} + \left[-F\left(2.94\frac{b^2}{a^2}+3.33\frac{a^2}{b^2}+0.89\right) +0.096\frac{\sigma b}{a}\right]f_{11} + \left[-F\left(3.36\frac{b^2}{a^2}+28\frac{a^2}{b^2}+2.88\right) +0.16\frac{\sigma b}{a}\right]f_{13}=0 \tag{18} \]

\[ \frac{dV_o}{df_{13}} = 2\left[-F\left(2.42\frac{b^2}{a^2}+204\frac{a^2}{b^2}+14.78\right)+0.115\frac{\sigma b}{a}\right]f_{13} + \left[-F\left(2.8\frac{b^2}{a^2}+2.01\frac{a^2}{b^2}+0.28\right) +0.07\frac{\sigma b}{a}\right]f_{11} + \left[-F\left(3.36\frac{b^2}{a^2}+28\frac{a^2}{b^2}+2.88\right) +0.16\frac{\sigma b}{a}\right]f_{12}=0 \tag{19} \]

Denoting in expression (17) the coefficients of \(f_{11}, f_{12}, f_{13}\) by \(A_1, A_2, A_3\), and in expressions (18) and (19), respectively, by \(B_1, B_2, B_3\) and \(C_1, C_2, C_3\), we can write the condition which determines the inequality to zero of \(f_{11}, f_{12}, f_{13}\) in the following form

\[ \begin{vmatrix} A_1 & A_2 & A_3\\ B_1 & B_2 & B_3\\ C_1 & C_2 & C_3 \end{vmatrix} =0 \]

or

\[ A_1[B_2C_3-C_2B_3]-A_2[B_1C_3-C_1B_3]+A_3[B_1C_2-C_1B_2]=0. \tag{20} \]

Solving equation 20 for various values of the ratio \(\frac{a}{b}\), we obtain a series of values of the critical stress. The results are summarized in Table 4.

Table 4.

The calculation we have carried out refers to the case in which, along the \(X\)-axis, the plate buckles in one half-wave.

Taking into account that, when buckling occurs in several half-waves, each of them may be regarded as an independent plate, it is easy to extend the results obtained to the case in which, during buckling of the plate along the \(X\)-axis, any number of half-waves is formed. This is done as follows.

Analyzing the graphical representation of the variation of \(\sigma_{cr}\) as a function of the ratio \(\frac{a}{b}\) for the case \(m=1\) (Fig. 3), we see that at first, with an increase in the ratio \(\frac{a}{b}\), \(\sigma_{cr}\) decreases, reaching its minimum at \(\frac{a}{b}=0.833\); then it begins to increase, and at \(\frac{a}{b}=1.2\) the value of \(\sigma_{cr}\) found under the assumption that the plate buckles in one half-wave proves to be equal to \(\sigma_{cr}\) for the ratio \(\frac{a}{b}=0.6\).

Fig. 3.

It follows that, for the above-mentioned ratio of the length of the plate to its width, it may have the first buckled form of equilibrium in the form of one or two half-waves.

Further, within a certain interval of the ratio \(\frac{a}{b}\), the value of \(\sigma_{cr}\) when the plate buckles in two half-waves proves to be less than when it is bent into one half-wave.

This indicates that one must use the curve \(m=2\), which is obtained from the curve \(m=1\) by doubling the abscissae.

In the same way, the curves \(m=3\), \(m=4\), etc., may be constructed.

The intersection of the curves \(m=1\), \(m=2\) determines the moment of transition from one half-wave to two.

Likewise, the intersection of the curves \(m=2\), \(m=3\) gives the value of the ratio \(\frac{a}{b}\) starting from which 3 half-waves are obtained, and so on.

The graph presented in Fig. 3 should also be used in determining the number of half-waves formed along the \(X\)-axis when the plate buckles, and the value of \(\sigma_{cr}\) occurring in this case. For a sufficiently long plate one may take

\[ \sigma_{cr} = 55 \frac{E}{1-\mu^2} \left(\frac{h}{b}\right)^2 \]

and, when it buckles, it is divided by nodal lines into segments with a length-to-width ratio equal to \(0.833\).

1. Skerblom, Course of Welded Structures. ↩

2. \[ D=\frac{E h^3}{12(1-\mu^2)} \]
   —cylindrical rigidity. ↩