Stability Analysis of a Mine Vertical Water Pipeline
Abstract Generated abstract
This paper develops an approximate stability calculation for a vertical mine water pipeline divided by shaft fastenings into separate sections. Treating the lowest section as the critical part of a weighty vertical rod, it applies an energy method with an asymmetric trigonometric form for the bent axis, accounting for both the weight of the pipe above the section and the section’s own weight. The derivation yields parametric relations between the dimensionless stability parameter and the critical section length, supplemented by tables and a diagram for practical interpolation, including cases with additional top loads. The paper also contrasts this approach with a simpler Euler calculation that neglects self-weight, showing that the rough method can give qualitatively different and sometimes fictitious results.
Full Text
Prof. G. V. TRAPEZNIKOV.
CALCULATION OF A VERTICAL MINE WATER PIPELINE FOR STABILITY
The vertical pipe of a mine water pipeline must have, along its length, a number of fastenings connecting it with the shaft framework; in the absence of such fastenings, even with the pipe installed strictly vertically, longitudinal buckling is inevitable. The insignificant resistance to longitudinal buckling is explained by the large ratio \(\frac{L}{i}\) for the pipe as a whole. By means of clamps, the entire length of the pipe \(L\) is divided into a number of sections, in each of which the above-mentioned ratio \(\frac{l}{i}\), where \(l\) is the length of the section under consideration, will be less than \(\frac{L}{i}\), thereby increasing the stability of the vertical pipe. In our practice, the calculation of the length of an individual section is reduced to the calculation of the lowest of them. In doing so, it is assumed that it may be regarded as a free column, with hinged ends, loaded only by a concentrated axial force equal to the weight of the entire overlying part of the pipe. As for foreign practice, it is reflected rather peculiarly in the catalogues of firms supplying mine water-supply pipes and fittings for them, which give no indications of a computational nature justifying one or another adopted distance from one fastening to another. Moreover, two different firms, supplying water pipes identical in dimensions, material, and installation, give distances between fastenings differing by a factor of 2–3. But since the length of an individual section enters Euler’s formula squared, such a discrepancy in the figures expressing the lengths of sections places a pipe fastened by one method, in comparison with a pipe fastened by another method, under conditions 4–9 times more favorable in terms of its stability.
In the present article an approximate solution is given to the problem of finding the minimum value of the dimensions of the sections, using the method of determining the equilibrium form of a bent rod under the condition of a minimum of the potential energy of its deformation under the prescribed conditions.
The problem, in the scheme (Fig. 1), is represented as follows: there is a vertical, straight, weighty rod of length \(L\). At a distance \(l\) from the lower, hinged-fixed end, it is fixed in such a way that, while at the point of fastening it has complete freedom along the vertical for rotation and displacement of the cross-section, at the same time it cannot move the latter in the horizontal plane. The lower segment, having length \(l\), is in the most unfavorable conditions with respect to the possibility of being bent by the action of a longitudinal load. On the one hand, it is subjected to the action of its own weight; on the other—to the action of a concentrated longitudinal load equal to the weight of that part of the rod which lies above this segment and has length equal to \(L-l\).
Drawing No. 1.
In addition to these loads, the rod is acted upon by a transverse force \(P\), which is the reaction of the fastening. When the upper end of the loaded segment settles, the latter moves perpendicular to the direction of the force \(P\), and therefore the work of the force \(P\) will be equal to zero.
The action of the rod’s own weight on a segment of length \(l\) is expressed in the fact that, in the presence of longitudinal bending, this part of the rod will not deform symmetrically. Therefore, for the approximate equation of the elastic line, it is necessary to take an asymmetric cyclic function:
\[ y=a\sin\frac{\pi x}{l}+b\sin\frac{2\pi x}{l}\ldots\ldots\ldots(1) \]
where \(a\) and \(b\) are constant parameters of the curve representing the bent axis of the rod.
For simplicity of notation in the derivations, we shall write:
\[ \frac{\pi x}{l}=\beta \]
and remember that:
\[ dx=\frac{l}{\pi}\,d\beta \]
and in substituting the limits we shall have:
\[ \begin{aligned} &\text{instead of } x=0 \text{ — } \beta=0,\\ &\text{instead of } x=l \text{ — } \beta=\pi. \end{aligned} \]
Thus:
\[ y=a\sin\beta+b\sin2\beta\ldots\ldots\ldots(2) \]
And the derivatives of the function with respect to \(x\):
\[ \frac{dy}{dx}=\frac{\pi}{l}(a\cos\beta+2b\cos2\beta)\ . . . . . . . . . (3) \]
\[ \frac{d^{2}y}{dx^{2}}=-\frac{\pi^{2}}{l^{2}}(a\sin\beta+4b\sin2\beta)\ . . . . . . . . (4) \]
The potential energy of the bent segment will be expressed as:
\[ U=\frac{1}{2EJ}\int_{0}^{l}M_x^2\,dx =\frac{EJ}{2}\int_{0}^{l}\left(\frac{d^2y}{dx^2}\right)^2dx= \]
\[ =\frac{\pi^3EJ}{2l^3}\int_{0}^{\pi} \left[a^2\sin^2\beta+8ab\sin\beta\cos2\beta+16b^2\sin^22\beta\right]\,d\beta= \]
\[ =\frac{\pi^3EJ}{2l^3} \left[ a^2\left(\frac{\beta}{2}-\frac{\sin2\beta}{4}\right) +8ab\left(\frac{\sin\beta}{2}-\frac{\sin3\beta}{6}\right) +\right. \]
\[ \left. +8b^2\left(\beta-\frac{\sin4\beta}{4}\right) \right]_{0}^{\pi} = \frac{\pi^4EJ}{4l^3}(a^2+16b^2)\ . . . . . (5) \]
This potential energy is stored by the segment under consideration as a result of the work performed by the external load, which includes both the weight of the rod in the part located above the given segment and the segment’s own weight. Let us determine the work of the named loads.
If the total weight of the rod is \(G\), then the weight of the upper part will be:
\[ G_{\text{upper}}=\frac{L-l}{L}G. \]
The work expended by the concentrated force \(\frac{L-l}{L}G\) will be expressed as:
\[ U_G=\frac{L-l}{2L}G\int_{0}^{l}\left(\frac{dy}{dx}\right)^2dx = \frac{\pi(L-l)}{2Ll}G\int_{0}^{\pi} (a^2\cos^2\beta+4ab\cos\beta\cos2\beta+ \]
\[ +4b^2\cos^22\beta)\,d\beta = \frac{\pi(L-l)}{2Ll}G \left[ a^2\left(\frac{\beta}{2}+\frac{\sin2\beta}{4}\right) +4ab\left(\frac{\sin\beta}{2}+\right. \]
\[ \left. +\frac{\sin3\beta}{6}\right) +2b^2\left(\beta+\frac{\sin4\beta}{4}\right) \right]_{0}^{\pi} = \frac{\pi^2(L-l)G}{4Ll}(a^2+4b^2)\ . . (6) \]
The work expended by the uniformly distributed load will be:
\[ U_q=\frac{q}{2}\int_0^l (l-x)\left(\frac{d^2y}{dx}\right)^2 dx = \]
\[ =\frac{q}{2}\int_0^\pi(\pi-\beta) \left(a^2\cos^2\beta+4ab\cos\beta\cos2\beta+4b^2\cos^2 2\beta\right)d\beta = \]
\[ =\frac{q}{2}\left[ \frac{\pi^2}{2}(a^2+4b^2) -\left|a^2\left\{\beta\left(\frac{\beta}{2}+\frac{\sin2\beta}{4}\right)-\frac{\beta^2}{4}+\frac{\cos2\beta}{8}\right\}\right| +\right. \]
\[ \left. +4ab\left\{\beta\left(\frac{\sin\beta}{2}+\frac{\sin3\beta}{6}\right) +\frac{\cos\beta}{2}+\frac{\cos3\beta}{18}\right\} +\right. \]
\[ \left. +b^2\left\{2\beta\left(\beta+\frac{\sin4\beta}{4}\right)-\beta^2+\frac{\cos4\beta}{8}\right\}\bigg|_0^\pi \right] = \]
\[ =\frac{\pi^2q}{8}\left(a^2+\frac{160}{9\pi^2}ab+4b^2\right). \]
Since:
\[ \frac{160}{9\pi^2}=1.806\approx\frac{9}{5} \]
and
\[ q=\frac{G}{L}, \]
then
\[ U_q=\frac{\pi^2G}{40L}(5a^2+9ab+20b^2). \]
Equating the potential energy of the bent rod to the work expended by the external loads:
\[ U=U_G+U_q \]
we obtain:
\[ \frac{\pi^4EJ}{4l^3}(a^2+16b^2) = \frac{\pi^2(L-l)G}{4Ll}(a^2+4b^2) + \frac{\pi^2G}{40L}(5a^2+9ab+20b^2). \]
Let us divide both sides of the equation by \(a^2\). Since this equation is homogeneous with respect to \(a\) and \(b\), ultimately we shall be dealing only with the ratio of \(b\) to \(a\), which we shall denote by:
\[ \frac{b}{a}=\alpha. \]
After a small transformation, our final equation will take the form:
\[ G_{\text{upper}}=q(L-l)=\frac{10\pi^2 EJ}{l^2}\cdot \frac{1+16\alpha^2}{10(1+4\alpha^2)+\dfrac{l}{L-l}(5+9\alpha+20\alpha^2)} \tag{2} \]
It is obvious that the weight of the upper part of the rod must have the smallest possible value.
Finding the magnitude of the lower section \(l\) is complicated, in our problem, by the fact that the equation obtained is nonhomogeneous with respect to \(l\) and \(L\). If the equation were homogeneous, we could seek the quantity \(\dfrac{l}{L}\), the change of which would be connected with the change of the parameter \(\alpha\). In the present case, however, every new value of \(L\) will correspond to a new length of the section \(l\), and, moreover, to a new value of the parameter \(\alpha\).
Therefore we shall temporarily modify the condition of the problem, which will make it possible to find an equation relating \(l\) and \(L\), and to determine the functional form of the parameter.
The new condition of the problem will be formulated as follows:
There is a rod of constant length \(l\), with hinged ends, the upper end being able to move freely in the vertical direction. It is necessary to determine what length a rod of the same material and cross-section will have which, when placed on the first rod, will put it under conditions of longitudinal buckling. In other words, the weight of the second rod must be the critical force for the first, already uniformly loaded by its own weight.
Let us denote the quantity:
\[ \frac{\pi^2 EJ}{l^2}=\Phi_l . \]
Equation (2) will be rewritten as:
\[ q(L-l)= \frac{10\Phi_l(1+16\alpha^2)-ql(5+9\alpha+20\alpha^2)} {10(1+4\alpha^2)} \quad \ldots (3) \]
In order that a rod of constant length \(l\) may be in a state of longitudinal buckling, it is necessary that the quantity \(L-l\), or, equivalently, \(G_{\text{upper}}\), should attain a minimum value.
Fulfilling this condition, we must solve the equation:
\[ \frac{dq(L-l)}{d\alpha}=0, \]
whence we obtain the equality, where \(l\) will already be the critical length:
\[ \Phi_l = q l_{\mathrm{cr}} \frac{3(1 - 4\alpha^2)}{80\alpha} \ldots \ldots \ldots \ldots \ldots \ldots (4) \]
If the function \(\alpha\) were of a simpler form, it would be better to determine \(\alpha\) through \(\Phi_l\) and substitute the value of \(\alpha\) thus obtained into the equation for \(L-l\). The relation equation \(L-l=\varphi(\Phi_l)\) would then be obtained at once. Carrying out such a substitution will lead to an equation which, owing to its complexity and cumbersomeness, will be almost unusable. To solve the problem, we shall choose the path of using the obtained functions in parametric form, i.e., without eliminating \(\alpha\).
Let us denote the ratio:
\[ \frac{l_{\mathrm{cr}}}{L}=\lambda_{\mathrm{cr}}. \]
Substitution of the value of \(\Phi_l\) into equation (3) gives:
\[ \lambda_{\mathrm{cr}}=\frac{l_{\mathrm{cr}}}{L}=\frac{8\alpha}{3+40\alpha-48\alpha^2}\ldots \ldots \ldots (5) \]
This will be one equation; the other we obtain as follows:
\[ \frac{\Phi_l}{q l_{\mathrm{cr}}} = \frac{\pi^2 E J}{q l_{\mathrm{cr}}^3} = \frac{\pi^2 E J}{q L^3}\left(\frac{L}{l_{\mathrm{cr}}}\right)^3 = \frac{\pi^2 E J}{G L^2 \lambda^3}. \]
Let us denote:
\[ \frac{\pi^2 E J}{G L^2}=\Phi \]
then:
\[ \Phi=\left(\frac{\Phi_l}{q l_{\mathrm{cr}}}\right)\lambda_{\mathrm{cr}}^3 = \frac{3(1-4\alpha^2)}{80\alpha}\lambda_{\mathrm{cr}}^3 \ldots \ldots \ldots (6) \]
Equations (5) and (6), given in parametric form, make it possible to determine \(\lambda_{\mathrm{cr}}\) for a known \(\Phi\), and conversely, if a table of values of \(\lambda_{\mathrm{cr}}\) and \(\Phi\) is compiled for varying \(\alpha\). Such a table is given below, as well as a diagram showing the character of the variation of the function \(\Phi\) as a function of \(\lambda_{\mathrm{cr}}\). When using the table, intermediate values of all quantities may be determined by linear interpolation, since the differences in the degrees are small (the exception is the first rows of the table, which have no special practical significance and are given here in order to provide an idea of the initial ordinates of the graph). The table gives \(\Delta\Phi\) and \(\Delta\lambda_{\mathrm{cr}}\) between consecutive values of the functions, to facilitate interpolation.
In the case when the length of the entire rod is comparatively small, it may turn out to be equal to or even less than its critical length. Let us determine the magnitude of the critical length when the latter is equal to the length of the entire rod.
In this case, obviously:
\[ \lambda_{\mathrm{cr}}=\frac{l_{\mathrm{cr}}}{L}=1 \quad\text{and}\quad \Phi_1=\frac{\pi^2 EJ}{L_{\mathrm{cr}}^2} \]
and equations (3) and (6) become the following:
\[ 0=\frac{10\pi^2 EJ}{L_{\mathrm{cr}}^2}(1+16\alpha^2)-qL_{\mathrm{cr}}(5+9\alpha+20\alpha^2) \]
and
\[ \frac{\pi^2 EJ}{qL_{\mathrm{cr}}^3}=\frac{3(1-4\alpha^2)}{80\alpha}. \]
Their simultaneous solution with respect to \(\alpha\) gives the equation:
\[ 192\alpha^4+160\alpha^3+24\alpha^2+40\alpha-3=0. \]
Hence we obtain the root that satisfies our condition:
\[ \alpha=0.0693. \]
Substitution of the obtained value of \(\alpha\) into equation (4) gives:
\[ \Phi_{1\mathrm{cr}}=0.5311\,qL_{\mathrm{cr}} \]
whence:
\[ (qL)_{\mathrm{cr}}=G_{\mathrm{cr}}=\frac{\pi^2 EJ}{(0.729L)^2} \]
i.e., the length coefficient in this case is:
\[ \psi=0.729. \]
Table of values of \(\Phi\) and \(\lambda_{\mathrm{cr}}\).
| \(\alpha\) | \(\lambda_{\mathrm{cr}}\) | \(\Delta\lambda_{\mathrm{cr}}\) | \(\Phi\) | \(\Delta\Phi\) | \(\alpha\) | \(\lambda_{\mathrm{cr}}\) | \(\Delta\lambda_{\mathrm{cr}}\) | \(\Phi\) | \(\Delta\Phi\) |
|---|---|---|---|---|---|---|---|---|---|
| 0,001 | 0,0263 | 0,0257 | 0,0007 | 0,0019 | 9 | 0,2145 | 0,0211 | 0,0411 | 0,0070 |
| 2 | 0,0520 | 0,0249 | 0,0026 | 0,0031 | 0,010 | 0,2356 | 0,0206 | 0,0490 | 0,0083 |
| 3 | 0,0769 | 0,0244 | 0,0057 | 0,0041 | 11 | 0,2562 | 0,0202 | 0,0573 | 0,0087 |
| 4 | 0,1013 | 0,0237 | 0,0097 | 0,0050 | 12 | 0,2764 | 0,0197 | 0,0660 | 0,0089 |
| 5 | 0,1250 | 0,0232 | 0,0147 | 0,0056 | 13 | 0,2961 | 0,0193 | 0,0749 | 0,0091 |
| 6 | 0,1482 | 0,0227 | 0,0203 | 0,0064 | 14 | 0,3154 | 0,0189 | 0,0840 | 0,0093 |
| 7 | 0,1709 | 0,0220 | 0,0267 | 0,0070 | 15 | 0,3343 | 0,0185 | 0,0931 | 0,0095 |
| 8 | 0,1929 | 0,0216 | 0,0337 | 0,0074 | 16 | 0,3528 | 0,0182 | 0,1029 | 0,0096 |
| $\alpha$ | $\lambda_{\mathrm{cr}.}$ | $\Delta\lambda_{\mathrm{cr}.}$ | $\Phi$ | $\Delta\Phi$ | $\alpha$ | $\lambda_{\mathrm{cr}.}$ | $\Delta\lambda_{\mathrm{cr}.}$ | $\Phi$ | $\Delta\Phi$ |
|---|---|---|---|---|---|---|---|---|---|
| 17 | 0,3710 | 0,0177 | 0,1125 | 0,0097 | 44 | 0,7542 | 0,0113 | 0,3628 | 0,0080 |
| 18 | 0,3887 | 0,0174 | 0,1222 | 0,0098 | 45 | 0,7655 | 0,0111 | 0,3708 | 0,0078 |
| 19 | 0,4061 | 0,0171 | 0,1320 | 0,0099 | 46 | 0,7766 | 0,0110 | 0,3786 | 0,0078 |
| 0,020 | 0,4232 | 0,0167 | 0,1419 | 0,0099 | 47 | 0,7876 | 0,0108 | 0,3864 | 0,0076 |
| 21 | 0,4399 | 0,0164 | 0,1518 | 0,0099 | 48 | 0,7984 | 0,0107 | 0,3940 | 0,0075 |
| 22 | 0,4563 | 0,0161 | 0,1617 | 0,0099 | 49 | 0,8091 | 0,0106 | 0,4015 | 0,0074 |
| 23 | 0,4724 | 0,0150 | 0,1716 | 0,0099 | 0,050 | 0,8197 | 0,0104 | 0,4089 | 0,0073 |
| 24 | 0,4883 | 0,0155 | 0,1815 | 0,0098 | 51 | 0,8301 | 0,0103 | 0,4162 | 0,0072 |
| 25 | 0,5038 | 0,0152 | 0,1913 | 0,0098 | 52 | 0,8404 | 0,0101 | 0,4234 | 0,0070 |
| 26 | 0,5190 | 0,0150 | 0,2011 | 0,0098 | 53 | 0,8505 | 0,0101 | 0,4304 | 0,0070 |
| 27 | 0,5340 | 0,0147 | 0,2109 | 0,0097 | 54 | 0,8606 | 0,0099 | 0,4374 | 0,0069 |
| 28 | 0,5487 | 0,0145 | 0,2206 | 0,0096 | 55 | 0,8705 | 0,0097 | 0,4443 | 0,0067 |
| 29 | 0,5632 | 0,0142 | 0,2302 | 0,0095 | 56 | 0,8802 | 0,0097 | 0,4510 | 0,0066 |
| 0,030 | 0,5774 | 0,0139 | 0,2397 | 0,0095 | 57 | 0,8899 | 0,0096 | 0,4576 | 0,0066 |
| 31 | 0,5913 | 0,0138 | 0,2492 | 0,0093 | 58 | 0,8995 | 0,0094 | 0,4642 | 0,0064 |
| 32 | 0,6051 | 0,0135 | 0,2585 | 0,0093 | 59 | 0,9089 | 0,0094 | 0,4706 | 0,0064 |
| 33 | 0,6186 | 0,0133 | 0,2678 | 0,0092 | 0,060 | 0,9183 | 0,0092 | 0,4770 | 0,0062 |
| 34 | 0,6319 | 0,0131 | 0,2770 | 0,0091 | 61 | 0,9275 | 0,0091 | 0,4832 | 0,0062 |
| 35 | 0,6450 | 0,0129 | 0,2861 | 0,0089 | 62 | 0,9366 | 0,0091 | 0,4894 | 0,0060 |
| 36 | 0,6579 | 0,0126 | 0,2950 | 0,0089 | 63 | 0,9457 | 0,0089 | 0,4954 | 0,0060 |
| 37 | 0,6705 | 0,0125 | 0,3039 | 0,0088 | 64 | 0,9546 | 0,0088 | 0,5014 | 0,0058 |
| 38 | 0,6830 | 0,0123 | 0,3127 | 0,0086 | 65 | 0,9634 | 0,0088 | 0,5072 | 0,0058 |
| 39 | 0,6953 | 0,0122 | 0,3213 | 0,0085 | 66 | 0,9722 | 0,0087 | 0,5130 | 0,0057 |
| 0,040 | 0,7075 | 0,0119 | 0,3298 | 0,0085 | 67 | 0,9809 | 0,0086 | 0,5187 | 0,0056 |
| 41 | 0,7194 | 0,0118 | 0,3383 | 0,0083 | 68 | 0,9894 | 0,0085 | 0,5243 | 0,0056 |
| 42 | 0,7312 | 0,0116 | 0,3466 | 0,0081 | 69 | 0,9979 | 0,5298 | ||
| 43 | 0,7428 | 0,0114 | 0,3547 | 0,0081 | 0,0693 | 1,0000 | 0,5311 |
Diagram \(\Phi\)—\(\lambda_{\mathrm{cr}}\).
Fig. No. 2.
In the diagram \(\Phi\)—\(\lambda_{\mathrm{cr}}\), the solid line gives the curve expressing the relation presented here in parametric form:
\[ \Phi=\frac{3(1-4a^2)}{80a}\lambda_{\mathrm{cr}}^2, \]
\[ \lambda_{\mathrm{cr}}=\frac{8a}{3+40a-48a^2}. \]
The dotted curve pertains to the rough approximate calculation discussed at the end of the article. It expresses the dependence:
\[ \Phi=\lambda^2(1-\lambda). \]
In order to obtain \(l_{\text{calc.}}\), we must use the relation
\[ n=\frac{P_{\text{calc.}}}{P_{\text{cr.}}}=\frac{l_{\text{cr.}}^2}{l_{\text{calc.}}^2}, \]
where \(n\) is the stability coefficient.
Hence:
\[ l_{\text{calc.}}=\frac{l_{\text{cr.}}}{\sqrt{n}}. \]
Let us apply the table obtained to determine the minimum dimensions of the sections between fastenings of vertical pipes of the standard assortment adopted in mine construction. The figures of critical lengths given below in the table have been calculated to an accuracy of \(0.1\ m\); moreover, when rounding the obtained value of \(l_{\text{cr.}}\) to \(0.1\), the lower figure was taken.
The table of critical lengths of the lower section of a mine water pipeline contains:
In the first cell.
The first line—\(D\), the inside diameter of the pipe in \(mm\).
The second line—\(a\), the wall thickness of the pipe in \(mm\).
The third line—the quantity \(\Phi L^3\cdot 10^{-2}=\frac{\pi^2 EJ}{q}\cdot 10^{-2}\) in \(m^3\).
In the second cell.
The quantities \(\Phi=\frac{\pi^2 EJ}{qL^3}\), calculated for various \(L\).
In the third cell.
The obtained quantities \(l_{\text{cr.}}\) in \(m\), for various \(L\).
From consideration of the figures \(l_{\text{cr.}}\) for mine water-pipeline pipes it is seen that, with a stability coefficient even equal to 9, we are able to set, for the lowest section of the pipe, the spacing:
\[ l_{\text{calc.}}=\frac{l_{\text{cr.}}}{\sqrt{9}}=\frac{l_{\text{cr.}}}{3}. \]
Thus, in the worst case, for cast-iron pipes, the fastenings should be placed no closer than \(13\ m\), and for steel pipes—no closer than \(12\ m\), from one another.
Table of critical lengths of the lower section of a vertical mine water pipeline.
Cast-Iron Pipes
| Quantity | \(L\), m | 100 | 125 | 150 | 200 | 250 | 300 |
|---|---|---|---|---|---|---|---|
| \(D\), mm | 100 | 125 | 150 | 200 | 250 | 300 | |
| \(a\), mm | 8,5 | 9,0 | 9,5 | 10.0 | 11,5 | 12,5 | |
| \(\Phi L^3 \cdot 10^{-2} m^3\) | 2030 | 3066 | 4340 | 7550 | 11650 | 16625 | |
| \(\Phi\) | 50 | 1,6240 | 2,4528 | 3,4720 | 6,0400 | 9,3200 | 13,3000 |
| \(\Phi\) | 100 | 0,2030 | 0,3066 | 0,4340 | 0,7550 | 1,1650 | 1,6625 |
| \(\Phi\) | 150 | 0,0601 | 0,0908 | 0,1286 | 0,2237 | 0,3452 | 0,4926 |
| \(l_{\mathrm{crit}}\) | 50 | critical length greater than the length of the water conduit | critical length greater than the length of the water conduit | critical length greater than the length of the water conduit | critical length greater than the length of the water conduit | critical length greater than the length of the water conduit | critical length greater than the length of the water conduit |
| \(l_{\mathrm{crit}}\) | 100 | 51,2 | 67,4 | 85,6 | greater than the length of the water conduit | greater than the length of the water conduit | greater than the length of the water conduit |
| \(l_{\mathrm{crit}}\) | 150 | 39,3 | 50,9 | 60,0 | 83,0 | 109,4 | 141,2 |
Steel Pipes
| Quantity | \(L\), m | 100 | 125 | 150 | 200 | 250 | 300 |
|---|---|---|---|---|---|---|---|
| \(D\), mm | 100 | 125 | 150 | 200 | 250 | 300 | |
| \(a\), mm | 3,75 | 4,0 | 4,5 | 6,5 | 7,0 | 7,5 | |
| \(\Phi L^3 \cdot 10^{-2} m^3\) | 3747 | 5790 | 8300 | 14830 | 22980 | 32870 | |
| \(\Phi\) | 50 | 2,9976 | 4,6320 | 6,6400 | 11,8640 | 18,3840 | 26,2960 |
| \(\Phi\) | 100 | 0,3747 | 0,5790 | 0,8300 | 1,4830 | 2,2980 | 3,2870 |
| \(\Phi\) | 150 | 0,1110 | 0,1716 | 0,2459 | 0,4394 | 0,6809 | 0,9739 |
| \(\Phi\) | 200 | 0,0468 | 0,0724 | 0,1038 | 0,1854 | 0,2873 | 0,4109 |
| \(\Phi\) | 250 | 0,0240 | 0,0371 | 0,0531 | 0,0949 | 0,1471 | 0,2104 |
| \(\Phi\) | 300 | 0,0139 | 0,0214 | 0,0307 | 0,0549 | 0,0851 | 0,1217 |
| \(l_{\mathrm{crit}}\) | 50 | critical length | critical length | critical length | critical length | critical length | critical length |
| \(l_{\mathrm{crit}}\) | 100 | 77,1 | greater than the length of the water conduit | greater than the length of the water conduit | greater than the length of the water conduit | greater than the length of the water conduit | greater than the length of the water conduit |
| \(l_{\mathrm{crit}}\) | 150 | 54,9 | 70,1 | 85,0 | 129,5 | greater than the length of the water conduit | greater than the length of the water conduit |
| \(l_{\mathrm{crit}}\) | 200 | 45,9 | 58,1 | 70,9 | 98,9 | 129,5 | 164,5 |
| \(l_{\mathrm{crit}}\) | 250 | 40,3 | 50,6 | 61,5 | 84,3 | 107,8 | 133,3 |
| \(l_{\mathrm{crit}}\) | 300 | 36,4 | 45,4 | 55,0 | 73,6 | 95,3 | 116,3 |
In the case where the upper end of the vertical pipe also carries an additional load, in the form of the weight of various parts of heavy fittings, etc., the calculation of \(l_{\mathrm{crit}}\) must be carried out as follows.
Let the vertical pipe have length \(L\) and weight \(G\). If the weight of the additional load is \(G_0\), then we can represent it as the weight of an added pipe of length \(L_0\), of the same cross-section and material.
Indeed:
\[ \frac{G}{L}=q=\frac{G_0}{L_0} \]
whence:
\[ L_0 = L \frac{G_0}{G}. \]
Then it may be assumed that we have a water pipeline with a pipe of length:
\[ L_1 = L + L_0 = L \left(1 + \frac{G_0}{G}\right) \]
and the quantity characterizing the degree of stability of the pipe is represented as:
\[ \Phi = \frac{\pi^2 EJ}{q (L + L_0)^3} = \frac{\pi^2 EJ \cdot G^3}{L^2 (G + G_0)^3} \]
Of some interest is the comparison of the calculation of \(l_{\mathrm{cr}}\), the lower section of the rod, taking into account the self-weight of the latter, with a calculation in which the self-weight of the section is neglected and it is assumed that the latter is loaded only by the weight of the overlying part of the rod.
When the self-weight of the lower section is neglected, we must apply Euler’s formula. Assuming, as before, both ends of the section to be hinged, we obtain:
\[ P_{\mathrm{cr}} = G_{\mathrm{top}} = q (L - l_{\mathrm{cr}}) = \frac{\pi^2 EJ}{l_{\mathrm{cr}}^2}. \]
Hence, having \(\dfrac{l}{L} = \lambda\):
\[ \Phi = \lambda^2 (1 - \lambda) \ldots \ldots \ldots \ldots \ldots \ldots (7) \]
Curve (7) is plotted by a dotted line on the graph given above.
The maximum value of \(\Phi\):
\[ \Phi_{\max} = 0.148. \]
Let us note that for \(\lambda = 0\) and \(\lambda = 1\), \(\Phi = 0\).
From consideration of such a rough approximate calculation and illustration of it by the graph, the following conclusions may be drawn:
1) If \(\lambda\) is gradually increased for a rod that has the stability characteristic:
\[ \Phi < 0.148 \]
then this increase may be carried out, within the limits of stability, up to \(\lambda\) determined by the smaller root of the equation:
\[ \lambda^2 (1 - \lambda) - \Phi = 0. \]
A further increase of \(\lambda\) is impossible, since it will entail loss of stability of the rod.
2) If it were possible, safely in the sense of preserving stability, to pass beyond this limit of \(\lambda\) and increase it further, then, upon \(\lambda\) reaching the value determined by the second, larger root of the equation given above, the rod would again acquire stability. Thus, in the approximate calculation, two different critical lengths of the lower section are obtained.
3) This is explained by the fact that, when the action of the lower section’s own weight is excluded from consideration, we may have two cases at the limit of stability. In the first case—a short rod loaded by the large weight of the part lying above it; and in the second—a long rod, for which the short part lying above provides a weight of sufficient magnitude for the latter to equal its critical force.
4) Rods having \(\Phi > 0.148\) will give fictitious values for \(\lambda\) and, therefore, in general, cannot be calculated by this method.
There is no need to give a more detailed analysis of the error obtained when applying the very approximate method of calculation. The curves of the graph \(\Phi - \lambda_{\mathrm{cr}}\) so clearly depict the quantitative and qualitative difference of the calculations that the presentation of numerical figures becomes superfluous.