On the Application of Electric Circuit Analysis Methods to Solving Bending Problems in Planar Rod Systems

G. E. Pukhov

Introduction

Methods for the study of electric circuits, especially in recent years, have come to be widely introduced into adjacent fields of science \[1—13 et al.\]. The most effective in this respect have proved to be the methods of electrical modeling and the well-known method of electromechanical analogies.

The present work has as its aim the description of a method for determining bending moments and displacements of planar rod systems (continuous beams, frames, etc.), based on the observed possibility of reducing a bending rod to an active electric three-terminal network.

For simplicity, only problems of statics are considered in the work.

1. Three-terminal electric equivalent circuit of a bending rod

Let the rod 1—2 (Fig. 1) be under the action of bending moments \(M_{12}\) and \(M_{21}\), reactions \(R_1\) and \(R_2\), and under the influence of an arbitrarily distributed external load along its length, the action of which is characterized by the area \((\omega)\) of the bending-moment diagram (under the condition of freely hinged supported ends) and by the distance \((ul)\) of its center of gravity from point 1 (or \(vl\) from point 2).

Fig. 1. Bending rod.

It is known \[14\] that in this case the relation between the angles of rotation \(\varphi_1\) and \(\varphi_2\) of the ends of the rod, the angle \(\psi\), characterizing the displacement \(y_2-y_1\) of the ends,

bending moments and, finally, support reactions is expressed by the system of equations:

\[ M_{12}=\frac{2EI}{l}(2\varphi_1+\varphi_2-3\psi)-\frac{2\omega}{l}(3v-1), \]

\[ -M_{21}=\frac{2EI}{l}(\varphi_1+2\varphi_2-3\psi)+\frac{2\omega}{l}(3u-1), \tag{1.1} \]

\[ R_1-R_{10}=-\frac{6EI}{l^2}(\varphi_1+\varphi_2-2\psi)-\frac{6\omega}{l^2}(u-v), \]

\[ R_2-R_{20}=\frac{6EI}{l^2}(\varphi_1+\varphi_2-2\psi)+\frac{6\omega}{l^2}(u-v) \]

(the action of longitudinal forces is not taken into account), where \(l\) is the length of the bar, \(R_{10}\) and \(R_{20}\) are the support reactions from the external load in a bar with free, hinged ends, \(I\) is the moment of inertia of the bar with respect to its axis, and \(E\) is the modulus of elasticity.

Since a bar is characterized, generally speaking, by four input quantities \((\varphi_1, M_{12}, y_1\) and \(R_1)\) and four output quantities \((\varphi_2, M_{21}, y_2\) and \(R_2)\), it is usually considered that the electrical representation of a bar is an eight-terminal network [8, 11].

The analysis of an electric circuit consisting of eight-terminal networks is complicated and leads to cumbersome computations. We shall show that a much simpler three-terminal circuit can be taken as the electrical circuit representing a bar.

Indeed, noting that

\[ M_{12}-M_{21}=-l(R_1-R_{10})=l(R_2-R_{20}), \tag{1.2} \]

and determining from the first two equations (1.3) the angles \(\varphi_1\) and \(\varphi_2\):

\[ \varphi_1=\frac{l}{3EI}M_{12}+\frac{l}{6EI}M_{21}+\frac{\omega v}{EI}+\psi, \tag{1.3} \]

\[ -\varphi_2=\frac{l}{6EI}M_{12}+\frac{l}{3EI}M_{21}+\frac{\omega u}{EI}-\psi \]

and, finally, treating the angles as voltages and electromotive forces, and the bending moments as currents of an electric circuit, it is not difficult to see that the relations (1.2) and (1.3) correspond to the circuit of an active electric three-terminal network shown in Fig. 2. This circuit\(^1\) is completely determined by the angle \(\psi\) and by the quantities

\[ r=\frac{l}{6EI}, \qquad \mathcal{E}_1=\frac{\omega v}{EI} \quad \text{and} \quad \mathcal{E}_2=\frac{\omega u}{EI}, \tag{1.4} \]

which we shall call the parameters of the replacement circuit of the bar.

Different loads of the bar and methods of fixing its ends are represented in the replacement circuit in the following way: 1) a bar with immov—

\(^1\) From a T-shaped replacement circuit it is easy, of course, if necessary, to pass to a \(\Pi\)-shaped circuit [18].

ends $\psi = 0$, 2) no external load is applied to the bar, $\omega = 0$ and, consequently, $\mathcal{E}_1 = \mathcal{E}_2 = 0$, 3) rigid fixing of the left end of the bar $\varphi_1 = 0$, 4) rigid fixing of the right end of the bar $\varphi_2 = 0$, 5) hinged fixing of the left end of the bar $M_{12} = 0$ (understood if no external bending moment is applied), etc.

Fig. 2. T-shaped substitution circuit (electrical analog) of a bending bar.

The values of the parameters $\mathcal{E}_1$ and $\mathcal{E}_2$ for various types of loading of the bar, calculated by formulas (1.4), are given in the appendix.

2. Procedure for calculating bar systems by the proposed method (the four-terminal method)

Having obtained above the three-terminal substitution circuit of a bending bar, one may propose the following procedure for calculating planar bar systems:

1) determination of parameters of the type $r$, $\mathcal{E}_1$ and $\mathcal{E}_2$ of the substitution circuits of the bars composing the system under consideration;

2) construction of the substitution circuit of the entire bar system, taking into account the boundary conditions (methods of fixing the ends, etc.);

3) calculation of the substitution circuit of the bar system as an ordinary electrical circuit, with recourse to solving the equations of statics ($\sum X = 0$, $\sum Y = 0$, etc.).

It should be noted that the substitution circuits of bar systems considered here belong to direct-current electrical circuits, whose calculations are the simplest. To determine the distribution of currents and voltages in the substitution circuits, any methods for analyzing linear electrical circuits may be applied, in particular the methods of loop currents and nodal voltages, the equivalent-generator theorem, the method of composite four-terminal networks [19—21], etc. For calculating circuits of especially complex configuration, it is expedient to apply the method of subcircuits, based on decomposing a complex circuit into successively, parallel, and mixed-connected subcircuits [22].

The bending moments acting at the various nodes of the bar system, and the angles of rotation of the nodes, will be represented in the substitution circuit by the corresponding electrical currents and voltages.

The further construction of diagrams of bending moments, transverse shearing forces, etc., may be carried out by the usual methods of structural mechanics.

Several examples explaining the foregoing are given in the appendix.

3. Remark on the equation of three moments

In calculations of continuous beams (Fig. 3a), the equation of three moments is often used:

\[ \frac{l_k}{I_k} M_{k-1} + 2\left( \frac{l_k}{I_k} + \frac{l_{k+1}}{I_{k+1}} \right) M_k + \frac{l_{k+1}}{I_{k+1}} M_{k+1} = \]

\[ = -6 \frac{\omega_k u_k}{I_k l_k^2} - 6 \frac{\omega_{k+1} v_{k+1}}{I_{k+1} l_{k+1}^2}. \tag{3.1} \]

Writing the loop-current equation for the \(k\)-th cell of the beam’s substitute circuit (Fig. 3b),

\[ r_k M_{k-1} + 2(r_k + r_{k+1})M_k + r_{k+1}M_{k+1} = -\mathcal{E}_2^{(k)} - \mathcal{E}_1^{(k+1)} \tag{3.2} \]

and taking expression (1.4) into account, we are convinced of the identity of equations (3.1) and (3.2).

Thus, the method of calculating continuous beams that uses the equation of three moments is an analogue of the calculation of chain electrical circuits by the loop-current method, which, let us note, is quite unsuitable for this purpose.

Fig. 3. Continuous beam and its substitute circuit.

4. Remark on the method of focal points

In structural mechanics there is a known theory of so-called focal ratios, which is of great importance for practical and theoretical calculations of continuous beams and frames. This theory reduces to finding, in each unloaded span, points (called moment foci) at which the envelope of bending moments changes sign. The theory is based on calculating the focal ratios \(\lambda\) from the formulas

\[ \lambda_k = -\frac{M_k}{M_{k-1}} = \frac{l_k - c_k}{l_k}, \tag{4.1} \]

where \(c_k\) is the distance from the focal point to the left support of the \(k\)-th span for a load located on spans to its right.

Fig. 4. Continuous beam and its substitute circuit for the case when the loads are to the right of the \(k\)-th support.

Turning to the substitute circuit (Fig. 4) of the \(k\)-th member and taking into account that both it and the substitute circuits of all left-hand spans are passive (parameters of the type \(\mathcal{E}_1\), \(\mathcal{E}_2\), and \(\psi\) are equal to zero), we see that the focal ratio is determined in the general case by the dependence:

\[ \lambda_k = -\frac{M_k}{M_{k-1}} = 2 + \frac{r_\Sigma}{r_k}, \tag{4.2} \]

where \(r_\Sigma\) is the resistance corresponding to the left part of the beam, except for the \(k\)-th span under consideration.

In special cases: a. \(r_{\Sigma}=0\) (the left end of the \(k\)-th span is fixed) \(\lambda_k=2\), b. \(r_{\Sigma}=\infty\) (the left end of the \(k\)-th span is hinged) \(\lambda_k=\infty\), etc. The latter relations are well known in structural mechanics. The value of \(r_{\Sigma}\) is easy to find directly from the beam substitution diagram by the usual methods of replacing series- and parallel-connected resistances by an equivalent resistance.

Thus, we arrive at the conclusion that there exists an analogy between the method of focal ratios of structural mechanics and the simplest techniques, known in electrical engineering, of the transformation (transfiguration) of electric circuits.

5. Conclusion

It was proved above that a bending rod can be reduced not only to an eight-terminal system, as is usually assumed, but also to a much simpler three-terminal system. Representing rods in the form of active electrical three-terminal networks makes it possible to somewhat simplify the calculations of plane rod systems and to apply to them any methods for investigating electric circuits.

The author hopes that the method presented will prove useful in the practice of calculating building structures, since in its application the capabilities of both the theory of electric circuits and the theory of structures can be used.

Of course, the proposed electrical substitution circuits for rod systems can easily be realized in the laboratory, after which the computation of bending moments and rotation angles is replaced by measurement of the corresponding currents and voltages.

Appendix 1. Parameters of the substitution circuit of a bending rod of constant cross section

Appendix 2. Examples of calculation of beams and frames.

Example 1. Determine the angle \(\varphi_1\) and the bending moment \(M_{21}\) for the beam in Fig. 5a. From the substitution circuit (Fig. 5b) we obtain

\[ M_{21}=-\frac{\mathcal{E}_2}{2r}=-\frac{q l^2}{8}, \tag{P.1} \]

\[ \varphi_1=(-r)(-M_{21})+\mathcal{E}_1=\frac{q l^3}{16EI}, \tag{P.2} \]

since

\[ r=\frac{l}{6EI} \quad \text{and} \quad \mathcal{E}_1=\mathcal{E}_2=\frac{q l^3}{24EI} \]

(see Appendix 1). The minus sign indicates that the fixing moment acts opposite to the chosen positive direction.

The values of \(M_{21}\) and \(\varphi_1\) are well known and are given, for example, in [15].

Fig. 5. Beam and its substitution circuit.

Fig. 6. Frame and its substitution circuit.

Example 2. Determine the bending moments at the nodes of the frame\(^1\), Fig. 6. Owing to the symmetric application of the load \(P\), the nodes of the frame will not be displaced, but will only rotate, and therefore \(\psi=0\). In the substitution circuit the electric voltages between the points \(0-a\) and \(0-b\) will obviously be equal to zero, as a result of which we immediately find

\[ M_{12}=M_{21} = -\frac{\mathcal{E}} {3r_{12}+3r_{14}+ \dfrac{(-r_{14})(3r_{11}+3r_{34})}{-r_{14}+3r_{11}+3r_{34}}} = \]

\[ = -\frac{\mathcal{E}} {3\left(r_{12}+\dfrac{3r_{14}^{2}}{2r_{11}+3r_{34}}\right)}. \tag{P.3} \]

Substituting here

\[ r_{12}=\frac{l_{12}}{6EI_{12}}=\frac{2}{3EI}, \qquad r_{14}=r_{23}=\frac{1}{EI}, \]

\[ r_{34}=\frac{4}{3EI}, \qquad \mathcal{E}=\frac{P l_{12}^{2}}{16EI_{12}}=\frac{2P}{EI}, \tag{P.4} \]

we finally obtain

\[ M_{12}=M_{21}=-\frac{12P}{23}=-0.5217P. \tag{P.5} \]

\(^1\) The data for the example are borrowed from [16, problem 13, 15].

The bending moments at nodes 3 and 4 will be

\[ M_{11}=M_{13}=M_{32}=M_{34}=-M_{12}\frac{r_{14}}{2r_{14}+3r_{31}} =-\frac{2P}{23}=-0.08695\,P. \tag{II.6} \]

The approximate solution of this problem by the method of foci, given in [16], has the form:

\[ M_{12}=M_{21}=-0.5224\,P,\qquad M_{11}=M_{13}=M_{32}=M_{34}=-0.087\,P. \tag{II.7} \]

Example 3. Let us find the support bending moments of the continuous beam\(^1\) in Fig. 7 due to a settlement of support \(B\) by the amount \(\Delta\).

Fig. 7. Continuous beam with settlement of a support and its replacement circuit

The parameters of the replacement circuit are equal to

\[ r=\frac{l}{6EI}\quad \text{and}\quad \psi=\frac{\Delta}{l}. \tag{II.8} \]

On the basis of the theorem on the equivalent generator (Thévenin—Helmholtz theorem), directly from the circuit we find the bending moment \(M_c\):

\[ M_c= -\frac{\psi-\dfrac{2\psi}{4r}\,5r} {\dfrac{5r(-r)}{5r+(-r)}+3r+\dfrac{5r(-r)}{5r+(-r)}} = \]

\[ =-\frac{3\psi}{7r} =-\frac{18}{7}\,\frac{EI}{l^2}\,\Delta =-2.57\,\frac{EI}{l^2}\,\Delta . \tag{II.9} \]

The bending moments in sections \(B\) and \(D\) will be

\[ M_B= \frac{\psi-\left(6r-\dfrac{5}{4}r\right)M_c}{5r} = \frac{85\psi}{140r} = \frac{51}{14}\,\frac{EI}{l^2}\,\Delta = 3.65\,\frac{EI}{l^2}\,\Delta, \]

\[ M_D= M_c\frac{(-r)}{4r} = \frac{3\psi}{28r} = \frac{9}{14}\,\frac{EI}{l^2}\,\Delta = 0.642\,\frac{EI}{l^2}\,\Delta. \tag{II.10} \]

\(^1\) The data are borrowed from [16, problem 10.17].

The found values \(M_B\), \(M_C\), and \(M_D\) coincide with those obtained by the usual methods of structural mechanics [16, problem 10.17].

Example 4. Determine the fixed-end moments at the joints of the frame of an intermediate floor slab (Fig. 8a).

Fig. 8. Frame of an intermediate floor slab and its equivalent circuit.

The equivalent circuit of the frame is shown in Fig. 8b.
The parameters \(\mathcal{E}\) and \(r\) of the circuit are equal to\(^1\)

\[ \mathcal{E}=\frac{q l^3}{24EI}=\frac{2\cdot 5^3}{24EI}=\frac{125}{12EI}, \qquad r=\frac{l}{6EI}=\frac{5}{6EI}. \tag{П.11} \]

Let us note that

\[ \frac{\mathcal{E}}{r}=12.5\ \text{t m}. \tag{П.12} \]

Calculating the circuit by any of the methods of electrical networks (it is most expedient to use the transfiguration method), we find

\[ \begin{aligned} M_{21}&=-\frac{11}{32}\frac{\mathcal{E}}{r}-\frac{\psi}{8r}, & M_{32}&=\frac{2}{32}\frac{\mathcal{E}}{r}-\frac{2\psi}{8r}, \\[4pt] M_{23}&=-\frac{6}{32}\frac{\mathcal{E}}{r}+\frac{2\psi}{8r}, & M_{31}&=\frac{1}{32}\frac{\mathcal{E}}{r}+\frac{\psi}{8r}, \\[4pt] M_{25}&=\frac{5}{32}\frac{\mathcal{E}}{r}+\frac{3\psi}{8r}, & M_{36}&=-\frac{1}{32}\frac{\mathcal{E}}{r}+\frac{3\psi}{8r}. \end{aligned} \tag{П.13} \]

The equation of statics is

\[ R_5+R_6-\frac{M_{25}+M_{36}}{h}=0. \tag{П.14} \]

\(^1\) The data of the example are borrowed from [17, p. 12].

allows one to determine the angle of rotation of the columns \(2\!-\!5\) and \(3\!-\!6\):

\[ \psi=-\frac{\mathrm{Э}}{6}=-\frac{125}{72EI}. \tag{P.15} \]

Substituting the values of \(\psi\) and \(r\) into equations (P.13), we finally obtain

\[ \begin{aligned} M_{21}&=-\frac{775}{192}=-4.036\ \text{t m},& M_{32}&=\frac{125}{96}=1.302\ \text{t m},\\ M_{23}&=-\frac{550}{192}=-2.864\ \text{t m},& M_{34}&=\frac{25}{192}=0.13\ \text{t m},\\ M_{25}&=\frac{225}{192}=1.172\ \text{t m},& M_{36}&=-\frac{225}{192}=-1.172\ \text{t m}. \end{aligned} \tag{P.16} \]

Exactly the same values of the bending moments are given in the work of G. V. Ulyaninsky [17, p. 121], from which the data of the example considered were borrowed.

Fig. 9. Two-story frame and its equivalent circuit.

Example 5. Determine the moment at joint No. 3 of the frame shown in Fig. 9a. The parameters of the equivalent circuit (Fig. 9b) are equal to

\[ r_{12}=\frac{l_{12}}{6EI_{12}}=\frac{1}{EI}=r,\qquad r_{25}=\frac{5}{6}r,\qquad r_{23}=r, \]

\[ r_{31}=\frac{10}{9}r,\qquad \mathrm{Э}=\frac{q l_{31}^{3}}{24EI_{31}}=\frac{5\cdot 10^{3}}{48}r. \tag{P.17} \]

By virtue of the symmetry of the system, the voltages between the points \(0\!-\!a\) and \(0\!-\!b\) are equal to zero. Therefore the required moment is found simply by Ohm’s law:

\[ M_{3}=-\frac{\mathrm{Э}}{r_{\mathrm{e}}} =-\frac{5\cdot 47\cdot 10^{3}}{16\cdot 704} \approx -20.7\ \text{t m}, \tag{P.18} \]

where \(r_{\mathrm{e}}=\dfrac{704r}{3\cdot 47}\) is the resistance measured between the terminals of the source of e.m.f. \(\mathrm{Э}\), provided that the points \(a\) and \(b\) are short-circuited with point \(O\).

It should be noted that the solution of this problem by the approximate method of Prof. B. N. Zhemochkin, given in [16, problem 13, 22, pp. 116 and 394], yielded the value \(M_{3}=20.8\ \text{t m}\).

References

1. A. A. Kharkevich. Theory of Electroacoustic Apparatus, Moscow, 1940.

2. A. A. Kharkevich. Theory of Transducers, Gosenergizdat, 1948.

3. A. A. Kharkevich. On the calculation of mechanical oscillatory systems, ZhTF, vol. IV, no. 6, 1934.

4. V. V. Furduyev. Electroacoustics. GTTI, 1948.

5. V. V. Furduyev. Reciprocity Theorems, Gostekhizdat, 1948.

6. G. Olson. Dynamical Analogies, GIIL, 1947.

7. L. I. Gutenmakher. Electrical Modeling (Electrointegrator), Publishing House of the Academy of Sciences of the USSR, 1943.

8. S. G. Mileikovskii. On the problem of reducing an oscillating rod to an eight-terminal system, ZhTF, vol. XI, no. 8, 1941.

9. N. M. Tetelbaum. Investigation of torsional vibrations of shafts of piston engines by means of electrical modeling, Publishing House of BIT, Moscow, 1948.

10. N. N. Andreev. On the calculation of mufflers for internal-combustion engines, ZhTF, vol. XVI, no. 6, 1946.

11. N. M. Tetelbaum. Electrical modeling of elastic systems in dynamic problems of structural mechanics. In: Increasing the Strength of Machine Parts, Publishing House of the Academy of Sciences of the USSR, 1949.

12. Yu. N. Grizodub. Application of the theory of passive four-terminal networks to the calculation of pressure-oscillation propagation in branched hydraulic systems of aircraft engines, Automation and Telemechanics, vol. XI, 1950.

13. A. L. Goflin. An electrical model of a beam resting on an elastic foundation, Electricity, no. 5, 1947.

14. N. M. Rabinovich. Structural Mechanics of Rod Systems, Stroiizdat, 1946.

15. A. N. Dinnik (ed.). Handbook of Technical Mechanics, Gostekhizdat, 1949.

16. N. L. Kuzmin, V. G. Rekach, G. I. Rosenblat. Collection of Problems in the Theory of Structures, Stroiizdat, 1956.

17. G. V. Ulyadinskii. Calculation of Frame Structures, Tomsk, 1931.

18. P. L. Kalantarov. Theory of Alternating Currents, Gosenergizdat, 1940.

19. E. V. Zelyakh. Fundamentals of the theory of four-terminal networks, Vestnik elektrotekhniki, nos. 3 and 7, 1931.

20. G. E. Pukhov. Equations of an active four-terminal network, Scientific Notes of LPI, issue VIII, Lviv, 1949.

21. G. E. Pukhov. Geometrical theory of circuits composed of four-terminal networks, Scientific Notes of LPI, issue X, Lviv, 1949.

22. G. E. Pukhov. Theory of the subcircuit method, Electricity, no. 8, 1952.