AC : TEACHING DEFLECTIONS OF BEAMS: COMPARISON OF ADVANTAGES OF METHOD OF MODEL FORMULAS VERSUS METHOD OF SUPERPOSITION

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1 C 11-91: TECHING DEFLECTIONS OF EMS: COMPRISON OF DVNTGES OF METHOD OF MODEL FORMULS VERSUS METHOD OF SUPERPOSITION Ing-Chang Jong, Universit of rkansas Ing-Chang Jong is a Professor of Mechanical Engineering at the Universit of rkansas. He received a SCE in 1961 from the National Taian Universit, an MSCE in 196 from South Dakota School of Mines and Technolog, and a Ph.D. in Theoretical and pplied Mechanics in 1965 from Northestern Universit. He and Dr. ruce G. Rogers coauthored the textbook Engineering Mechanics: Statics and Dnamics, Oxford Universit Press (1991). Professor Jong as Chair of the Mechanics Division, SEE, , and received the rchie Higdon Distinguished Educator ard in 9. His research interests are in mechanics and engineering education. William T. Springer, Universit of rkansas William T. Springer is 1st Centur Chair in Mechanical Engineering and ssociate Professor at the Universit of rkansas. He received his SME in 197 from the Universit of Texas at rlington, his MSME in 1979 from the Universit of Texas at rlington, and his Ph.D. in Mechanical Engineering in 198 from the Universit of Texas at rlington. Dr. Springer is active in SME here he received the Dedicated Service ard in 6, as elected to Fello Grade in 8, and as aarded the S. Y. Zamrik Pressure Vessels and Piping Medal in 11. c merican Societ for Engineering Education, 11

2 Teaching Deflections of eams: Comparison of dvantages of Method of Model Formulas versus Method of Superposition bstract The method of model formulas is a ne method for solving staticall indeterminate reactions and deflections of elastic beams. Since its publication in the IJEE in 9, instructors of Mechanics of Materials have considerable interest in teaching this method to enrich students set of skills in determining beam reactions and deflections. esides, instructors are interested in seeing relative advantages of this method versus the traditional method of superposition. This paper is aimed at comparing the method of model formulas ith the method of superposition regarding (a) their methodolog and pedagog, (b) the availabilit of a one-page excerpt from the method of model formulas, (c) the availabilit of a one-page collection of deflection formulas of selected beams for the method of superposition, and (d ) assessment of their effectiveness in solving problems of reactions and deflections of beams in several identical given problems. I. Introduction eams are longitudinal members subjected to transverse loads. Students usuall first learn the design of beams for strength. Then the learn the determination of deflections of beams under a variet of loads. Methods used in determining staticall indeterminate reactions and deflections of elastic beams include: -1 method of integration (ith or ithout use of singularit functions), method using moment-area theorems, method of conjugate beam, method using Castigliano s theorem, method of superposition, method of segments, and method of model formulas. The method of model formulas 1 is a nel propounded method. eginning ith an elastic beam under a selected preset general loading, a set of four model formulas are derived and established for use in this ne method. These four formulas are expressed in terms of the folloing: (a) flexural rigidit of the beam; (b) slopes, deflections, shear forces, and bending moments at both ends of the beam; (c) tpical applied loads (concentrated force, concentrated moment, linearl distributed force, and uniforml distributed moment) somehere on the beam. For starters, one must note that a orking proficienc in the rudiments of singularit functions is a prerequisite to using the method of model formulas. To benefit a ider readership, hich ma have different specialties in mechanics, and to avoid or minimize an possible misunderstanding, this paper includes summaries of the rudiments of singularit functions and the sign conventions for beams. Readers, ho are familiar ith these topics, ma skip the summaries. n excerpt from the method of model formulas is needed and shon in Fig. 1, courtes of IJEE. 1 esides, a collection of slope and deflection formulas of selected beams for the method of superposition is needed and shon in Fig., courtes of a textbook b S. Timoshenko and G. H. MacCullough.

3 Excerpt from the Method of Model Formulas Courtes: Int. J. Engng. Ed., Vol. 5, No. 1, pp. 65-7, 9 (a) (b) Positive directions of forces, moments, slopes, and deflections Va M a 1 P K a + x + x < x xp>+ < x x x x K> < > EI EI EI EI < x x > + < x u > + < x u > EI x EI x m m + < xxm> < xu m> EI EI ( u ) ( u ) (1) V M K P + x+ x + x < x x + < x x > x x EI EI EI < x x > + < x u > + < x u > 1 EI ( ) m m + < x xm > < x um> a a a a P> < K > ( u x) EI 1EI u x () Va L Ma L P ( L xp) K ( L xk ) ( L x) b a + EI + EI EI + EI 6 EI ( L x ) + ( L u ) + EI ( Lu ) m m + ( Lxm) ( Lum) EI EI ( ux) EI ( ux) () VL a MaL P K b a + al+ + ( L x ) + ( Lx ) ( Lx ) EI EI EI ( L x) + ( L u ) + ( Lu ) 1 EI( u x) EI 1EI( u x) m m + ( Lxm) ( Lum) P K () Fig. 1. Model loading and beam deflection formulas for the method of model formulas

4 Deflection Formulas of Selected eams for the Method of Superposition Courtes: Timoshenko and MacCullough, Elements of Strength of Materials, th Ed., pp , D. Van Nostrand Compan, Inc., 199 Fig.. Slope and deflection formulas of selected beams for the method of superposition

5 Summar of rudiments of singularit functions Notice that the argument of a singularit function is enclosed b angle brackets (i.e., < >). The argument of a regular function continues to be enclosed b parentheses [i.e., ( )]. The rudiments of singularit functions include the folloing: 1,8,9 n n < x a > ( xa) if xa and n> (5) n < x a> 1 if xa and n (6) n < x a > if x a< or n< (7) x n 1 n+ 1 < x a > dx x a if n n + 1 < > > (8) x < x a> n dx < x a> n + 1 n if (9) d n n1 < x a > n< x a > dx if n> (1) d n n1 < x a > < x a > dx if n (11) Equations (6) and (7) impl that, in using singularit functions for beams, e take b b 1 for b (1) for b< (1) Summar of sign conventions for beams In the method of model formulas, the adopted sign conventions for various model loadings on the beam and for deflections of the beam ith a constant flexural rigidit EI are illustrated in Fig. 1. Notice the folloing ke points: shear force is positive if it acts upard on the left (or donard on the right) face of the beam element [e.g., V a at the left end a, and V b at the right end b in Fig. 1(a)]. t ends of the beam, a moment is positive if it tends to cause compression in the top fiber of the beam [e.g., M a at the left end a, and M b at the right end b in Fig. 1(a)]. If not at ends of the beam, a moment is positive if it tends to cause compression in the top fiber of the beam just to the right of the position here it acts [e.g., the concentrated moment K K and the uniforml distributed moment ith intensit m in Fig. 1(a)]. concentrated force or a distributed force applied to the beam is positive if it is directed donard [e.g., the concentrated force P P, the linearl distributed force ith intensit on the left side and intensit 1 on the right side in Fig. 1(a), here the distribution becomes uniform if 1]. The slopes and deflections of a beam displaced from to ab are shon in Fig. 1(b). Note that positive slope is a counterclockise angular displacement [e.g., a and b in Fig. 1(b)]. positive deflection is an upard linear displacement [e.g., a and b in Fig. 1(b)].

6 Methodolog and pedagog of the method of model formulas The four model formulas in Eqs. (1) through () ere derived in great detail in the paper that propounded the method of model formulas. 1 For convenience of readers, let us take a brief overvie of ho these model formulas are obtained. asicall, it starts out ith the loading function q, ritten in terms of singularit functions for the beam ab in Fig. 1; as follos: q V < x> + M < x> P< x x > + K< x x > < x x > < x x > + < x u > + < x u > u x u x 1 1 a a P K m < x x > m < x u > 1 1 m m integrating q, one can rite the shear force V and the bending moment M for the beam ab in Fig. 1. Letting the flexural rigidit of the beam ab be EI, be the deflection, be the slope, and be the second derivative of ith respect to the abscissa x, hich defines the position of the section along the axis of the beam under consideration, one ma appl the relation EI M and readil obtain the expressions for EI and EI via integration. The slope and deflection of the beam are a and a at its left end a (i.e., at x ), and are b and b at the right end b (i.e., at x L), as illustrated in Fig. 1. Imposition of these boundar conditions ill ield the four model formulas in Eqs. (1) through (). Note that L in the model formulas in Eqs. (1) through () is a parameter representing the total length of the beam segment. In other ords, this L is to be replaced b the total length of the beam segment to hich the model formulas are applied. Furthermore, notice that this method allos one to treat reactions at interior supports (i.e., those not at the ends of the beam) as applied concentrated forces or moments, as appropriate. ll one has to do is to simpl impose the additional boundar conditions at the points of interior supports for the beam segment. Thus, staticall indeterminate reactions as ell as slopes and deflections of beams can be determined. beam needs to be divided into segments for analsis onl if (a) it is a combined beam (e.g., a Gerber beam) having discontinuities in slope at hinge connections beteen segments, and (b) it contains segments ith different flexural rigidities (e.g., a stepped beam). Methodolog and pedagog of the method of superposition The method of superposition for the deflection of beams is a traditional method that can be found in most textbooks on mechanics of materials. -8 The methodolog and pedagog of this method ma not require a detailed description in this paper. asicall, this method requires that a table containing a good collection of slope and deflection formulas of selected beams, such as the one shon in Fig., be available. In this method, the resulting deflection of a beam due to the various loads applied on the beam is taken to be the same as the sum of the deflections of the beam due to individual loads applied one at a time on the beam. Hoever, this method ill fail to ork if the table lacks formulas for certain tpes of load (e.g., concentrated moment acting at other than an end point of a simpl supported beam). (1)

7 II. Teaching and Learning a Ne Method via Contrast beteen Solutions Mechanics is mostl a deductive science, but learning is mostl an inductive process. For the purposes of teaching and learning, all examples ill first be solved b the ne method of model formulas (MoMF). Then the same problems in the examples ill be solved b the traditional method of superposition (MoS) unless impossible using just the formulas for slopes and deflections of beams that are shon in Fig.. To sta ithin a limit in this paper, formulas that are available from other sources ill not be admitted in the solution using MoS in the examples. Example 1. simpl supported beam D ith constant flexural rigidit EI and total length L is acted on b a concentrated force P at and a concentrated moment PL at C as shon in Fig.. Determine (a) the slopes and D at and D, respectivel; (b) the deflection at. Fig.. Simpl supported beam D carring concentrated loads Solution. The beam is staticall determinate. Its free-bod diagram is shon in Fig.. Fig.. Free-bod diagram of the simpl supported beam D Using MoMF: In appling the method of model formulas to this beam, e must adhere to the sign conventions as illustrated in Fig. 1. t the left end, the moment M is, the shear force V is 5P/, the deflection is, but the slope is unknon. t the right end D, the deflection D is, but the slope D is unknon. Note in the model formulas that e have xp L/ for the concentrated force P at and xk L/ for the concentrated moment PL at C. ppling the model formulas in Eqs. () and (), successivel, to this beam D, e rite 5 / ( P ) L P L PL L D + + L + L EI EI EI 5 P/ L PL L ( ) P L + L + + L L EI These to simultaneous equations ield 1 17 PL PL D 81EI 1 Using the value of and appling the model formula in Eq. (), e rite L 5 P/ L PL + L/

8 We report that 1PL 81EI 17PL D 1 PL 8 Using MoS: The beam in Fig. carries a concentrated moment PL at C, hich is not at the end of the beam. Since the table of formulas in Fig. lacks this tpe of load applied at a point other than the end of the simpl supported beam, the problem in this example is prevented from being solved b the method of superposition and the use of just formulas in Fig.. ssessment of effectiveness. In this example, e see that the method of model formulas enables one to directl and successfull obtain the solutions. The method of superposition could not be emploed to solve the problem in this example solel because there are no formulas in Fig. for the slope and deflection of a simpl supported beam acted on b a concentrated moment at a point other than the end of the beam. This shos a donside of the MoS hen formulas in the list are inadequate. Thus, the MoMF appears to be more general and effective than the MoS. Example. cantilever beam C ith constant flexural rigidit EI and total length L is loaded ith a distributed load of intensit in segment as shon in Fig. 5. Determine (a) the slope and deflection at, (b) the slope and deflection at. Fig. 5. Cantilever beam C loaded ith a distributed load Solution. The beam is staticall determinate. Its free-bod diagram is shon in Fig. 6. Fig. 6. Free-bod diagram of the cantilever beam C Using MoMF: In appling the method of model formulas to solve the problem, e note that the shear force V and the bending moment M at the free end, as ell as the slope C and the deflection C at the fixed end C, are all zero. Seeing that the uniforml distributed load has x and u L/, e appl the model formulas in Eqs. () and () to the entire beam to rite L L + L + + L + L L + L + + These to simultaneous equations ield EI EI

9 7L 8EI 1L 8EI Using these values and appling the model formulas in Eqs. (1) and (), respectivel, e rite L L L/ EI L L 7L x L/ EI 19EI We report that 7L 8EI 1L 8EI L 8EI 7L 19EI Using MoS: For appling the method of superposition and adapting to the formulas in Fig., e ma turn the original given beam in Fig. 5 about a vertical axis through 18 and note that it is equivalent to the superposition (or sum) of to differentl loaded beams as shon in Fig Fig. 7. Original given beam is equivalent to the superposition of to beams shon Referring to the formulas for the rd beam in Fig. and appling the MoS to the beam, e rite L ( L/) 7L 8EI 7L 8EI L ( L/) L ( L/) 1L 1L + + 8EI 8EI 8EI 8EI Referring to the formulas for the rd beam in Fig. again, e rite d δ ( x 6L x Lx ) EI x L x Lx dx + + ( ) ( L /) L ( x + L x Lx ) + 8EI L/ ( L /) 7L ( x + 6L x Lx ) + EI 8EI 19EI L/ L 8EI 7L 19EI ssessment of effectiveness. In this example, e see that the MoMF enables us to directl and successfull obtain the solutions, hile the MoS requires some rotations and combinations of beams to eventuall arrive at the same solutions. The MoMF is more straightforard than the MoS; but the are about equall effective in solving the problem in this example.

10 Example. cantilever beam C ith constant flexural rigidit EI and total length L is propped at and carries a concentrated moment M at as shon in Fig. 8. Determine (a) the vertical reaction force and slope at, (b) the slope and deflection at. Fig. 8. Cantilever beam C propped at and carring a concentrated moment at Solution. The free-bod diagram of the beam is shon in Fig. 9, here e note that the beam is staticall indeterminate to the first degree. Fig. 9. Free-bod diagram of the propped cantilever beam C Using MoMF: In appling the method of model formulas to this beam, e first note that this beam has a total length of L, hich ill be the value for the parameter L in all of the model formulas in Eqs. (1) through (). We also note that the deflection C and the slope C at C, as ell as the deflection at, are all equal to zero. ppling the model formulas in Eqs. () and () to this beam, e rite ( L) M ( L L ) EI EI ( L) EI M + ( L) ( LL) These to simultaneous equations ield 9M 16L M L 8EI Using these values and appling the model formulas in Eqs. (1) and (), respectivel, e rite 5ML + L x L EI EI ML + L+ L x L EI We report that 9M 16L ML 8 EI 5ML EI ML EI

11 Using MoS: For appling the method of superposition and adapting to the formulas in Fig., e ma turn the original given beam in Fig. 8 about a vertical axis through 18 and note that it is equivalent to the superposition (or sum) of to differentl loaded beams as shon in Fig Fig. 1. Original given beam is equivalent to the superposition of to beams shon Referring to the formulas for the 1 st and 5 th beams in Fig., appling the MoS, and imposing the boundar condition of zero deflection at, e rite ML ML ( L) + L + EI EI EI ML ( L) ML EI EI 8EI 9M 16L ML d x ML L 5ML [ ( L) x] EI dx EI EI EI ML x ML 5L ML + [ ( L) x] + EI EI EI L We report that 9M 16L L ML 8 EI 5ML EI ML EI ssessment of effectiveness. In this example, e see that the MoMF enables us to directl and successfull obtain the solutions, hile the MoS requires some rotations and combinations of beams to eventuall arrive at the same solutions. The MoMF is more straightforard than the MoS; but the are about equall effective in solving the problem in this example. Example. continuous beam C ith constant flexural rigidit EI and total length L has a roller support at, a roller support at, and a fixed support at C. This beam carries a linearl distributed load and is shon in Fig. 11. Determine (a) the vertical reaction force and slope at, (b) the vertical reaction force and slope at. Fig. 11. Continuous beam C carring a linearl distributed load

12 Solution. The free-bod diagram of the beam is shon in Fig. 1. We readil note that the beam is staticall indeterminate to the second degree. Fig. 1. Free-bod diagram of the continuous beam C Using MoMF: In appling the method of model formulas to this beam, e notice that the beam C has a total length L, hich ill be the value for the parameter L in all model formulas in Eqs. (1) through (). We see that the shear force V at left end is equal to, the moment M and deflection at are zero, the deflection at is zero, and the slope C and deflection C at C are zero. ppling the model formulas in Eqs. () and () to the beam C and using Eq. () to impose the condition that L ( ) at, in that order, e rite L ( ) / ( ) / + + L L + L ( L) EI EI EIL ( /) + ( L L) + ( L L) + 6 EI EIL ( ) ( ) L / ( ) + ( L) + + ( L L) + ( L) ( L) EI 1 EIL ( /) 5 + ( L L) + ( L L) + EI 1 EIL ( ) / 5 / ( /) EI 1 EIL 5 + L + L + + L L These three simultaneous equations ield 9 L L 1 L 1 1EI 56 Using these values and appling the model formula in Eq. (1), e rite / ( /) EI 6 EI EI L L + L + + L L L 168EI We report that 9L 1 L 1EI 1L 56 L 168EI Using MoS: For appling the method of superposition and adapting to the formulas in Fig., e ma turn the original given beam in Fig. 1 about a vertical axis through 18 and note that it is equivalent to the superposition (or sum) of six differentl loaded beams as shon in Fig. 1.

13 Fig. 1. Original given beam is equivalent to the superposition of six beams as shon We have assumed that the reactions at and are and, respectivel. Referring to the formulas for the 1 st, nd, rd, and th beams in Fig. and appling the MoS, e impose the to boundar conditions at and at, successivel, to rite and ( /)( L) ( L) L L ( /) L ( /) L + + L + L 8EI EI 8EI + EI EI ( L) L + + [ ( L) L] EI ( /) x x x 6( L) ( L) x 1( L) 1( L) x 5( L) x x EI + 1( L) EI + L L L [ ( L) x] + 8EI EI EI L L ( /) x L The above to equations ma be simplified and shon to be equivalent to the matrix equation 6 89L 1 5L Solving the matrix equation, e obtain and report that 9L 1 9L 1 1L 56 1L 56 Using the obtained values for and, referring to the formulas for the 1 st, nd, rd, and th beams in Fig., and appling the MoS, e rite ( ) ( /)( L) ( L) L ( /) L L L L + EI EI EI EI 1EI

14 d ( /) x + dx EI x 6( L) ( Lx ) d x + 1( L) 1( L) x+ 5( Lx ) x dx 1( L) EI L ( /) L d x L [ ( L) x] EI dx EI L 168EI L L L We report that L 1EI L 168EI ssessment of effectiveness. In this example, e see that the method of model formulas enables us to directl and successfull obtain the solutions, hile the method of superposition requires some rotations and combinations of beams to eventuall arrive at the same solutions. For the problem in this example, the road to the final solution is more straightforard hen e use the MoMF, but it is more meandering and complex hen e use the MoS. Clearl, the MoMF is more effective and has an edge over the MoS in this case. III. Effective Teaching of the MoMF The method of model formulas is a general methodolog that emplos a set of four equations to serve as model formulas in solving problems involving staticall indeterminate reactions, as ell as slopes and deflections, of elastic beams. The first to model formulas are for the slope and deflection at an position x of the beam and contain rudimentar singularit functions, hile the other to model formulas contain onl traditional algebraic expressions. Generall, this method is more direct in solving beam deflection problems. Most students favor this method because the can solve problems in shorter time using this method and the score higher in tests. The examples in Section II provide a variet of head-to-head comparisons beteen solutions b the method of model formulas and those b the traditional method of superposition. one-page excerpt from the method of model formulas, such as that shon in Fig. 1, must be available to those ho used this method. Past experience shos that the folloing steps form a pedagog that can be used to effectivel introduce and teach the method of model formulas to students to enrich their stud and set of skills in finding staticall indeterminate reactions and deflections of elastic beams in the undergraduate course of Mechanics of Materials: First, teach the fundamental method of integration and the imposition of boundar conditions. Teach the rudiments of singularit functions and utilize them in the method of integration. Go over briefl the derivation 1 of the four model formulas in terms of singularit functions.

15 Give students the heads-up on the folloing features in the method of model formulas: No need to integrate or evaluate constants of integration. Not prone to generate a large number of simultaneous equations even if the beam carries multiple concentrated loads (forces or moments), the beam has one or more simple supports not at its ends, the beam has linearl distributed loads not starting at its left end, the beam has linearl distributed loads not ending at its right end, and the beam has non-uniform flexural rigidit but can be divided into uniform segments. Demonstrate solutions of several beam problems b the method of model formulas. Check the solutions obtained (e.g., comparing ith solutions b another method). In a 9-student Mechanics of Materials class (for sophomores and juniors) in the spring semester of 1, the students ere first taught the method of integration (MoI), then the method of model formulas (MoMF) in the stud of deflection of beams. ased on available data: (a) hen the students learn and use the MoI to solve staticall indeterminate reactions and deflections of beams in a quiz, the results of their performance ere 1 s, s, 7 C s, 1 D s, and 18 F s, ith overall class average equal to 69.1%; (b) hen these same students learn and use the MoMF to solve staticall indeterminate reactions and deflections of beams in a quiz, the results of their performance ere 5 s, 18 s, C s, 1 D s, and 9 F s, ith overall class average equal to 79.7%. The grade distributions for these to quizzes are sho in Fig. 1. The MoMF has appeared to be a more accessible method to the students in understanding the deflection of beams, and students favor this method in solving the problems. (a) Quiz result on using MoI (b) Quiz result on using MoMF (overall average 69.1%) (overall average 79.7%) Fig. 1. Students performance in using MoMF IV. Concluding Remarks In the method of model formulas, no explicit integration or differentiation is involved in appling an of the model formulas. The model formulas essentiall serve to provide material equations (hich involve and reflect the material propert) besides the equations of static equilibrium of

16 the beam that can readil be ritten. Selected model applied loads are illustrated in Fig. 1(a), hich cover most of the loads encountered in undergraduate Mechanics of Materials. In the case of a nonlinearl distributed load on the beam, the model formulas ma be modified b the user for such a load. The method of model formulas is relativel ne; it is best taught to students as an additional or alternative method after the have first learned the method of integration. -1 This ne method is found to be more general and effective than the traditional method of superposition, as shon in Section II. Learning and using this ne method ill enrich students stud and set of skills in determining reactions and deflections of beams. Moreover, this ne method provides engineers ith a means to independentl check their solutions obtained using traditional methods. References 1. I. C. Jong, n lternative pproach to Finding eam Reactions and Deflections: Method of Model Formulas, International Journal of Engineering Education, Vol. 5, No. 1, pp. 65-7, 9.. S. Timoshenko and G. H. MacCullough, Elements of Strength of Materials (rd Edition), Van Nostrand Compan, Inc., Ne York, NY, S. H. Crandall, C. D. Norman, and T. J. Lardner, n Introduction to the Mechanics of Solids (nd Edition), McGra-Hill, Ne York, NY, R. J. Roark and W. C. Young, Formulas for Stress and Strain (5th Edition), McGra-Hill, Ne York, NY, F. L. Singer and. Ptel, Strength of Materials (th Edition), Harper & Ro, Ne York, NY, Ptel and J. Kiusalaas, Mechanics of Materials, rooks/cole, Pacific Grove, C,. 7. J. M. Gere, Mechanics of Materials (6th Edition), rooks/cole, Pacific Grove, C,. 8. F. P. eer, E. R. Johnston, Jr., J. T. DeWolf, and D. F. Mazurek, Mechanics of Materials (5th Edition), McGra- Hill, Ne York, NY, R. G. udnas and J. K. Nisbett, Shigle s Mechanical Engineering Design (8th Edition), McGra-Hill, Ne York, NY, H. M. Westergaard, Deflections of eams b the Conjugate eam Method, Journal of the Western Societ of Engineers, Vol. XXVI, No. 11, pp , H. T. Grandin, and J. J. Rencis, Ne pproach to Solve eam Deflection Problems Using the Method of Segments, Proceedings of the 6 SEE nnual conference & Exposition, Chicago, IL, I. C. Jong, Deflection of a eam in Neutral Equilibrium à la Conjugate eam Method: Use of Support, Not oundar, Conditions, 7 th SEE Global Colloquium on Engineering Education, Cape Ton, South frica, October 19-, I. C. Jong, Determining Deflections of Elastic eams: What Can the Conjugate eam Method Do That ll Others Cannot? International Journal of Engineering Education, Vol. 6, No. 6, pp. 1-17, 1.