THE USE OF KIRCHOFF S CURRENT LAW AND CUT-SET EQUATIONS IN THE ANALYSIS OF BRIDGES AND TRUSSES

Transcription

1 Session TH US O KIRCHO S CURRNT LAW AND CUT-ST QUATIONS IN TH ANALYSIS O BRIDGS AND TRUSSS Ravi P. Ramachanran an V. Ramachanran. Department of lectrical an Computer ngineering, Rowan University, Glassboro, New Jersey, 0808, U.S.A.. Department of lectrical an Computer ngineering, Concoria University, Montreal, Quebec, CANA., HG M8. Abstract - The purpose of this paper is to show that the analysis of trusses (an hence that of briges) can be effectively carrie out using the three concepts of Basic lectric Circuit Analysis, namely, the Superposition Theorem, Kirchhoff s Current Law an the Cut-set metho. The curricular effect of this stuy is the improvement of multiisciplinary engineering eucation by relating the sophomore courses of Statics an Circuits an putting the courses uner one common analysis framework. Introuction In any engineering curriculum, it is common practice to teach Statics an Basic Circuit Analysis in the sophomore year as separate subjects. In the subject of Statics [], the analysis of briges an trusses is taught using the two concepts base on equilibrium equations, namely (i) the algebraic sum of moments taken at a point is zero, an (ii) the algebraic sum of the various forces at any joint in each of the vertical an horizontal irections will be equal to zero. In Basic Circuit Analysis [], the subject matter starts with Kirchoff s Current Law (KCL). or any network, KCL states that the vectorial sum of the various currents incient at any noe is always equal to zero. Kirchhoff s Voltage Law (KVL) is not consiere in this paper. In aition, when sinusoial excitation is consiere, such a current can be represente as a phasor. Also, since linear networks are consiere, the principle of Superposition hols. The Superposition theorem states that the total response in a branch is the vectorial sum of the various responses, each response being obtaine when only source is consiere, with all other inepenent sources being mae equal to zero. In aition to the above, the cut-set concept is also iscusse. The cut-set concept is the generalization of the KCL at a noe in that the KCL hols for a surface also []. Page..

2 It is the purpose of this paper to show that the above-mentione concepts of basic circuit analysis can be effectively use in the analysis of briges an trusses. This common framework in the analysis of structures an circuits shoul be shown to the stuents to provie a better comprehension of the relationship between the two subjects of Statics an Circuits. or purposes of illustration, the example in [, pp.84] will be consiere in this paper with the ifference that the length of each arm is arbitrary an the forces acting at junctions are inepenent of one another. This is as shown in igure. f f A θ B θ θ θ4 θ D 4 igure A general truss consiere for illustration purposes. Superposition Theorem The general analysis requires that the moment about C is taken first. This gives f ( + ) + f -. = 0. () The objective is to etermine the vertical component. It is reaily mae up of two parts (ue to f ) an (ue to f ). By making f = 0, we get ( ) = f + C () By making f = 0, we get Thus, = f () f ( + ) + f = + = (4) This concept can be reaily extene for any number of forces present. This permits us to etermine the contribution of each force to. The above iscussion makes it clear that the principle of superposition (which is the starting point an the basis for linear system theory [4]) can be reaily applie here. This means that one force at a time can be consiere, making all other forces equal to zero. This will Page..

3 enable one to etermine the contribution of each force to the reactions at the hinge ens by the equilibrium equation. In aition, the concept of scaling can be also be applie in that first, a unit force is applie an its effect can be etermine. Then, the require multiplication factor is applie to obtain the actual effect. This can be illustrate as follows: Let f = 0 an a unit force be applie at the junction A. Then, from (), we get ( ) = + a () When a force f is applie, this is multiplie by a factor f so that f( + ) = f. a = () The factor a is the influence coefficient. The effect of f can be obtaine in a similar fashion. Kirchhoff s Current Law (KCL) In orer to illustrate this principle, we choose one suitable junction for the purposes of analysis. In this case, we start with the junction A, which in circuit theory terms, is a noe. The corresponing force iagram, epicting both external an internal forces, is shown in igure. = + a f AB igure The force iagram for the junction A. Analysis of the force iagram yiels the equilibrium equation, + AB f = = (7) from which AB an are etermine given f. To use KCL, we make use of a funamental system moeling concept that states that a force an current are analogous through variables [4][]. This enforces the fact that the concept of the vectorial sum of forces at a joint being zero is the application of KCL to a Page..

4 structure. Continuing with the example, we construct a vector phasor iagram which is ientical to the shape of the truss at junction A. This is as shown in igure. f θ AB igure The vector or phasor iagram corresponing to junction A. This iagram is mae up of the various forces in each arm incient at the junction A. The forces in each arm are represente by an appropriate complex quantity. The horizontal axis becomes the real axis an the vertical axis becomes the imaginary axis. The irections of AB an can be chosen in an arbitrary manner initially. Application of KCL at noe A gives -jθ - jf + + e AB = 0 (8) where j is a complex number (j = -). While writing quation (8), the force (or equivalently, current) entering the junction (or equivalently, noe) is taken to be negative an the force leaving the junction is taken to be positive. quating the real parts, we get AB + cos(θ ) = 0 (0) Since we get cos( θ ) =, (0a) + + AB = (0b) quation (0b) shows that the irection of one of the forces, namely either AB or has to be reverse. quating the imaginary parts, we get -jf - j sin(θ ) = 0 (a) Since we get sin( θ ) =, (b) + Page..4

5 + f = () This clearly shows that the irection of has to be reverse, because the irection of f is alreay given. The combination of quations (0b) an () gives quation (7). This emonstrates clearly that KCL can be effectively use in the analysis. It is note that this analysis gives both the magnitue an the irection of the force in each arm. It is reaily seen that the horizontal component of the force is given by the real part an the vertical component of the force is given by the imaginary part. Cut-set quations It is known that every noe is a cut-set, in that the cut-set ivies the network graph into two istinct parts []. Some of the cut-sets (otte lines) are shown in igure 4. f f C A C B C C D igure 4 Some cut-sets in the graph epicting the truss. The equation corresponing to noe A is given in quation (8). This also gives the equation corresponing to the cut-set C. A similar equation can be written corresponing to noe D, which is as follows: jθ e jθ + + e 4 =0 D quation () is also the cut-set equation corresponing to C. quations (8) an () give the various forces along the branches, D, AB an DB. Also, aition of (8) an () yiels DB jθ - jf + + e 4 + = 0 AB DB This is the equation corresponing to the cut-set C. quation (4) can either be use to verify the various forces alreay etermine or use in the analysis to etermine the forces. As an illustration, equating the imaginary parts in (4), we get DB sin(θ 4 ) = f () Since D () (4) Page..

6 sin( θ ) = 4 ( ) + -, () we get ( - ) f. + DB = (7) This suffices to prove that the cut-set equations can be use effectively in the analysis. More cutsets can be use. In fact, Cut-set analysis gives the same equations as the Metho of Sections. These two methos are hence, analogous. Similar analysis can be carrie out for other noes, thereby completely etermining the various forces an their respective irections in all the branches. Conclusions The foregoing iscussion clearly shows that the three concepts of basic circuit analysis, namely Superposition, KCL an Cut-set analysis can be effectively use in the analysis of trusses. Thus, this paper makes a thrust at multiisciplinary engineering eucation by showing the equivalence of structures an circuits thereby epicting the isomorphism in the analysis techniques. It is suggeste that courses on Statics an Circuits can be improve an looke at uner one common analysis framework. References.. P. Beer an. R. Johnston, Jr., Vector Mechanics for ngineers: Statics, McGraw- Hill Book Co., 99.. J. W. Nilsson an S. A. Rieel, lectric Circuits, Aison -Wesley Book Co., 99.. N. Balabanian an T. Bickart, Linear Network Theory: Analysis, Properties, Matrix Publishers, R. C. Dorf an R. H. Bishop, Moern Control Systems, Aison-Wesley Book Co., J.. Linsay an V. Ramachanran, Moeling an Analysis of Linear Physical Systems, Weber Systems Inc., January 99. Acknowlegement The authors thank Dr. Rama Bhat for helpful iscussions uring the preparation of the paper. Page..

7 Biography Ravi P. Ramachanran is an Associate Professor in the Department of lectrical an Computer ngineering at Rowan University. He receive his Ph.D. from McGill University in 990 an has worke at AT&T Bell Laboratories an Rutgers University prior to joining Rowan. V. Ramachanran is a Professor in the Department of lectrical an Computer ngineering at Concoria University. He receive his Ph.D. from the Inian Institute of Science in 9 an was on the faculty at the Technical University of Nova Scotia prior to joining Concoria. Biography Page..7