AC : TEACHING DEFLECTIONS OF BEAMS: ADVANTAGES OF METHOD OF MODEL FORMULAS VERSUS THOSE OF CONJU- GATE BEAM METHOD

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1 C 1-998: TECHING DEFECTIONS OF EMS: DVNTGES OF METHOD OF MODE FORMUS VERSUS THOSE OF CONJU- GTE EM METHOD Dr. Ing-Chang Jong, Universit of rkansas Ing-Chang Jong is a professor of mehanial engineering at the Universit of rkansas. He reeived a.s.c.e. in 1961 from the National Taian Universit, a M.S.C.E. in 196 from South Dakota Shool of Mines and Tehnolog, and a Ph.D. in theoretial and applied mehanis in 1965 from Northestern Universit. He and rue G. Rogers oauthored the textbook Engineering Mehanis: Statis and Dnamis, Oxford Universit Press (1991). Jong as Chair of the Mehanis Division, SEE, , and reeived the rhie Higdon Distinguished Eduator ard in 9. His researh interests are in mehanis and engineering eduation. merian Soiet for Engineering Eduation, 1 Page

2 Teahing Defletions of eams: dvantages of Method of Model Formulas versus Those of Conjugate eam Method bstrat The method of model formulas is a reentl published method. It emplos a general model loading diagram and derived four ke equations as model formulas. These formulas an aount for the beam s flexural rigidit, applied onentrated loads, linearl distributed loads, and the boundar or support onditions. No expliit integration is needed in using the model formulas in this method. This method an be applied to solve most problems involving beam reations and defletions enountered in the teahing and learning of mehanis of materials. On the other hand, the onjugate beam method is a natural extension of the moment-area theorems. It is an elegant, effiient, and poerful method propounded b Westergaard in 191. Elementar presentation of this method did appear in some earl textbooks of mehanis of materials.,5 For reasons unknon, this method is urrentl missing in most suh textbooks. This paper is aimed at providing omparisons of the method of model formulas versus the onjugate beam method regarding their (a) pedagog and methodolog, (b) effetiveness in solving problems of defletions of beams and statiall indeterminate reations at supports via several head-to-head ontrasting solutions of the same problems, and () as to effetivel introdue and teah either of these methods to students. I. Introdution eams are longitudinal members subjeted to transverse loads. Students usuall first learn the design of beams for strength. Then the learn the determination of defletions of beams under loads. Methods used in determining statiall indeterminate reations and defletions of elasti beams inlude: 1-1 method of integration (ith or ithout use of singularit funtions), momentarea theorems, Castigliano s theorem, method of superposition, method of segments, method of model formulas, and onjugate beam method. The method of model formulas (MoMF) 1 is nel propounded in 9. set of four model formulas are derived and established for use in this ne method. The formulas are expressed in terms of the folloing: (a) flexural rigidit of the beam; (b) slopes, defletions, shear fores, and bending moments at both ends of the beam; () tpial applied loads (onentrated fore, onentrated moment, linearl distributed fore, and uniforml distributed moment) somehere on the beam. To use the MoMF, one must have rudiments of singularit funtions at pla and utilized an exerpt from this method as shon in Fig. 1, ourtes of IJEE. 1 This paper inludes a one-page of summaries of the rudiments of singularit funtions and the sign onventions for beams. Readers, ho are familiar ith these topis, ma skip the summaries. Page 5.17.

3 Exerpt from the Method of Model Formulas Courtes: Int. J. Engng. Ed., Vol. 5, No. 1, pp. 65-7, 9 (a) (b) Positive diretions of fores, moments, slopes, and defletions 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 Ma P ( xp) K ( xk ) ( x) b a + EI + EI EI + EI 6 EI ( x ) + ( u ) + EI ( u ) m m + ( xm) ( um) EI EI ( ux) EI ( ux) () V a Ma P K b a + a+ + ( x ) + ( x ) ( x ) EI EI EI ( x) + ( u ) + ( u ) 1 EI( u x) EI 1EI( u x) m m + ( xm) ( um) P K Fig. 1. Model loading and beam defletion formulas for the method of model formulas () Page 5.17.

4 Summar of rudiments of singularit funtions Consistent ith the ommonl used notations, the argument of a singularit funtion is enlosed b angle brakets (i.e., < >). The argument of a regular funtion ontinues to be enlosed b parentheses [i.e., ( )]. The rudiments of singularit funtions inlude the folloing: 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 funtions for beams, e take b b 1 for b (1) for b< (1) Summar of sign onventions for beams In the method of model formulas, the adopted sign onventions for various model loadings on the beam and for defletions of the beam ith a onstant flexural rigidit EI are illustrated in Fig. 1. Notie the folloing ke points: 8,9 shear fore is positive if it ats upard on the left (or donard on the right) fae 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 ause ompression 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 ause ompression in the top fiber of the beam just to the right of the position here it ats [e.g., the onentrated moment K K and the uniforml distributed moment ith intensit m in Fig. 1(a)]. onentrated fore or a distributed fore applied to the beam is positive if it is direted donard [e.g., the onentrated fore P P, the linearl distributed fore ith intensit on the left side and intensit 1 on the right side in Fig. 1(a), here the distribution beomes uniform if 1]. The slopes and defletions of a beam displaed from to ab are shon in Fig. 1(b). Note that positive slope is a ounterlokise angular displaement [e.g., a and b in Fig. 1(b)]. positive defletion is an upard linear displaement [e.g., a and b in Fig. 1(b)]. Page 5.17.

5 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 MoMF. 1 For onveniene of readers, let us take a brief overvie of ho these model formulas are obtained. asiall, it starts out ith the loading funtion q, 9 ritten in terms of singularit funtions 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 (1) integrating q, one an rite the shear fore V and the bending moment M for the beam ab in Fig. 1. etting the flexural rigidit of the beam ab be EI, be the defletion, be the slope, and be the seond derivative of ith respet to the absissa x, hih defines the position of the setion under onsideration along the axis of the beam, one ma appl the relation EI M and readil obtain the expressions for EI and EI via integration. The slope and defletion 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 ), as illustrated in Fig. 1. Imposition of these boundar onditions ill ield the four model formulas in Eqs. (1) through (). The pedagog of the MoMF lies in teahing and appling the four model formulas in this method. Note that in the model formulas in Eqs. (1) through () is a parameter representing the total length of the beam segment. In other ords, this is to be replaed b the total length of the beam segment to hih the model formulas are applied. Furthermore, notie that this method allos one to treat reations at interior supports (i.e., those not at the ends of the beam) as applied onentrated fores or moments, as appropriate. ll one has to do is to simpl impose the additional boundar onditions at the points of interior supports for the beam segment b using Eqs. (1) and (). Thus, statiall indeterminate reations as ell as slopes and defletions of beams an be determined. beam needs to be divided into segments for analsis onl if (a) it is a ombined beam (e.g., a Gerber beam) having disontinuities in slope at hinge onnetions beteen segments, and (b) it ontains segments ith different flexural rigidities (e.g., a stepped beam). Methodolog and pedagog of the onjugate beam method The onjugate beam method (CM) propounded b Westergaard 1 is a great method and is onsistent ith the moment-area theorems. The support onditions (free end, fixed end, simple support at the end of the beam, simple support not at the end of the beam, and unsupported hinge), rather than the traditional boundar onditions, are heavil used in this method. Earlier textbooks,5 inluded onl brief and elementar overage of this method. Someho most urrent prevailing textbooks drop the overage of this method. The pedagog of the CM lies in teahing and appling the ten guiding rules snthesized for this method. For onveniene of readers, these rules are listed belo, ourtes of IJEE. 1 Page

6 Guiding rules in the onjugate beam method: 1 Rule 1: The onjugate beam and the given beam are of the same length. Rule : The load on the onjugate beam is the elasti eight, hih is the bending moment M in the given beam divided b the flexural rigidit EI of the given beam. (This elasti eight is taken to at upard if the bending moment is positive to ause top fiber in ompression in beam onvention.) For eah existing support ondition of the given beam, there is a orresponding support ondition for the onjugate beam. The orrespondene is given b rules through 7 as follos: Existing support ondition in the given beam: Corresponding support ondition in the onjugate beam: Rule : Fixed end Free end Rule : Free end Fixed end Rule 5: Simple support at the end Simple support at the end Rule 6: Simple support not at the end Unsupported hinge Rule 7: Unsupported hinge Simple support Rule 8: The onjugate beam is in stati equilibrium. Rule 9: The slope of the given beam at an ross setion is given b the shear fore at that ross setion of the onjugate beam. (This slope is positive, or ounterlokise, if the shear fore is positive tending to rotate the beam element lokise in beam onvention.) Rule 1: The defletion of the given beam at an point is given b the bending moment at that point of the onjugate beam. (This defletion is upard if the bending moment is positive tending to ause the top fiber in ompression in beam onvention.) II. Teahing and earning Ne Methods via Contrast beteen Solutions Mehanis is mostl a dedutive siene, but learning is mostl an indutive proess. For the purposes of teahing and learning, all examples ill first be solved b the method of model formulas (MoMF). Then the same problems in the examples ill be solved b the onjugate beam method (CM). For onvenient omparison of effetiveness in the solutions b different methods, problems in previous examples 1 ill be emploed in illustrating solutions b the MoMF and the CM. beam is in neutral equilibrium if the fore sstem ating on the beam is statiall balaned and the potential energ of the beam in the neighborhood of its equilibrium onfiguration is onstant. This is illustrated in Fig.. Fig.. eam in neutral equilibrium on a simple support. Readers are advised to note that all methods, exept the CM, annot solve the tpe of problems involving defletions of beams in neutral equilibrium. In this regard, the CM is more general and poerful! To kno more about this feature, readers ma see the paper b Jong. 1 Page

7 Example 1. simpl supported beam D ith onstant flexural rigidit EI and total length is ated on b a onentrated fore P at and a onentrated moment P at C as shon in Fig.. Determine (a) the slopes and D at and D, respetivel; (b) the defletion at. Fig.. Given beam D, simpl supported and arring onentrated loads Solution. The beam is statiall determinate. Its free-bod diagram is shon in Fig.. Fig.. Free-bod diagram of the given beam D Using MoMF: In appling the method of model formulas to this beam, e must adhere to the sign onventions as illustrated in Fig. 1. t the left end, the moment M is, the shear fore V is 5P/, the defletion is, but the slope is unknon. t the right end D, the defletion D is, but the slope D is unknon. Note in the model formulas that e have xp / for the onentrated fore P at and xk / for the onentrated moment P at C. ppling the model formulas in Eqs. () and (), suessivel, to this beam D, e rite 5 / ( P ) P P D EI EI EI 5 P/ P ( ) P EI These to simultaneous equations ield 1 17 P P D 81EI 1 Using the value of and appling the model formula in Eq. (), e rite 5 P/ P + x / We report that 1P 81EI P 17P D 1 P 8 Using CM: In using the onjugate beam method to solve the problem in this example, e first make use of the free-bod diagram in Fig. and appl guiding rules 1,, and 5 in the CM, as listed at the end of Set I, to onstrut the onjugate beam for the given beam as shon in Fig. 5. Note that the bending-moment diagram is dran b parts from both ends toard. Page

8 Fig. 5. Conjugate beam for given beam D Fig. 6. FD of onjugate beam D The free-bod diagram of the onjugate beam D is shon in Fig. 6, here the reations at the simple supports and D are assumed to be ating donard; i.e., and D D. Notie that e have used a supersript on the smbols for these reations to signif that the are assoiated ith the onjugate beam, not the given beam. Next, referring to the onjugate beam in Fig. 6 and appling guiding rule 8 in the CM, e rite + Σ : + M D Σ F : P P P 9 9EI 9 6 9EI EI 5P P P + D 6 9EI EI 9EI These to simultaneous equations ield 1 P 17 D P 81EI 1 ording to the sign onventions for beams summarized in Set. I, the shear fores at and D in the onjugate beam in Fig. 6 are 1 V 17 P V P D D 81EI 1 ppling guiding rule 9 in the CM, e have V and V. We report that 1P 81EI P D 17P D 1 ppling guiding rule 1 in the CM, e rite M P P P P 9 6 9EI 8 EI 8 We report that P 8 D ssessment of effetiveness. In this example, e see that the method of model formulas enables one to diretl rite the pertinent equations and solve them to obtain the solutions. The onjugate beam method does not require the use of an exerpt of the model formulas. Hoever, the CM requires the appliation of the guiding rules to first onstrut the onjugate beam for the given beam, then rite the equations of equilibrium from the free-bod diagram for the Page

9 onjugate beam, solve the equations, and appl guiding rules 9 and 1 in the CM to get the slopes and defletions of the beam. In this example, e observe that the MoMF involves mostl algebrai ork from the use of the model formulas in the solution, hile the CM involves guiding rules, more geometr, more statis, and about the same amount of algebrai ork. oth MoMF and CM ield the same solutions and are equall effetive in solving the problem in this example. (Nevertheless, given an opportunit to hoose beteen these to methods to solve a beam defletion problem in the final exam of the author s lass MEEG 1 Mehanis of Materials, in fall 11, about 75% of the students prefer to use the MoMF.) Example. antilever beam C ith onstant flexural rigidit EI and total length is loaded ith a distributed load of intensit in segment as shon in Fig. 7. Determine (a) the slope and defletion at, (b) the slope and defletion at. Fig. 7. Cantilever beam C loaded ith a distributed load Solution. The beam is statiall determinate. Its free-bod diagram is shon in Fig. 8. Fig. 8. Free-bod diagram of the antilever beam C Using MoMF: In appling the method of model formulas to solve the problem, e note that the shear fore V and the bending moment M at the free end, as ell as the slope C and the defletion C at the fixed end C, are all zero. Seeing that the uniforml distributed load has x and u /, e appl the model formulas in Eqs. () and () to the entire beam to rite These to simultaneous equations ield EI EI 7 8EI 1 8EI Using these values and appling the model formulas in Eqs. (1) and (), respetivel, e rite Page

10 x / EI 7 x / EI 19EI We report that 7 8EI 1 8EI 8EI 7 19EI Using CM: In using the onjugate beam method to solve the problem in this example, e first make use of the free-bod diagram in Fig. 8 and appl guiding rules 1,,, and in the CM, as listed at the end of Set I, to onstrut the onjugate beam for the given beam C as shon in Fig. 9. Notie that the free end at in the given beam in Fig. 7 beomes a fixed end at in the onjugate beam in Fig. 9. The free-bod diagram of the onjugate beam C is shon in Fig. 1, here the unknon shear fore and the bending moment at the fixed end are assumed to at in the positive diretions in the beam onventions. Fig. 9. Conjugate beam for given beam C Fig. 1. FD of onjugate beam C Next, referring to Fig. 1 and appling guiding rule 8 in the CM, e rite + Σ F : + Σ : M 1 6 8EI + 8EI 8EI M EI 8EI 6 8EI These to simultaneous equations ield 7 1 M 8EI 8EI ppling guiding rules 9 and 1 in the CM, e have V, M. We report that 7 8EI 1 8EI Referring to Fig. 1, e find that the shear fore and bending moment at of the onjugate beam are Page

11 V 6 8EI 8EI M 7 M EI 19EI ppling guiding rules 9 and 1 in the CM, e have V and M. We report that 8EI 7 19EI ssessment of effetiveness. gain, e see that the method of model formulas enables one to diretl rite the pertinent equations and solve them to obtain the solutions. The onjugate beam method does not require the use of an exerpt of the model formulas. Hoever, the CM requires the appliation of the guiding rules to first onstrut the onjugate beam for the given beam, then rite the equations of equilibrium from the free-bod diagram for the onjugate beam, solve the equations, and appl guiding rules 9 and 1 in the CM to get the slopes and defletions of the beam. oth MoMF and CM ield the same solutions and are equall effetive in solving the problem in this example. Example. antilever beam C ith onstant flexural rigidit EI and total length is propped at and arries a onentrated moment M at as shon in Fig. 11. Determine (a) the vertial reation fore and slope at, (b) the slope and defletion at. Fig. 11. Cantilever beam C propped at and arring a onentrated moment at Solution. The free-bod diagram of the beam is shon in Fig. 1, here e note that the beam is statiall indeterminate to the first degree. Fig. 1. Free-bod diagram of the propped antilever 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, hih ill be the value for the parameter in all of the model formulas in Eqs. (1) through (). We also note that the defletion C and the slope C at C, as ell as the defletion at, are all equal to zero. ppling the model formulas in Eqs. () and () to this beam, e rite Page

12 ( ) M ( ) EI EI ( ) EI M + ( ) ( ) These to simultaneous equations ield 9M 16 M 8EI Using these values and appling the model formulas in Eqs. (1) and (), respetivel, e rite 5M x EI EI We report that M x EI 9M 16 M 8 EI 5M EI M EI Using CM: In using the onjugate beam method to solve the problem in this example, e first make use of the free-bod diagram in Fig. 1 and appl guiding rules 1,,, and 5 in the CM, as listed at the end of Set I, to onstrut the onjugate beam for the given beam C as shon in Fig. 1. Notie that the simple support at the end in the given beam C in Fig. 11 remains a simple support at the end in the onjugate beam C in Fig. 1; hoever, the fixed end at C in the given beam beomes a free end at C in the onjugate beam. The free-bod diagram of the onjugate beam C is shon in Fig. 1, here the unknon shear fore at the end is assumed to at in the positive diretion in the beam onventions. Fig. 1. Conjugate beam for given beam C Fig. 1. FD of onjugate beam C Next, referring to Fig. 1 and appling guiding rule 8 in the CM, e rite M EI EI + Σ M : Page

13 The above to simultaneous equations ield + M EI EI + Σ F : 9M 16 M 8EI ppling guiding rule 9 in the CM, e have V. We report that 9M 16 M 8 EI Referring to Fig. 1, e find that the shear fore and bending moment at of the onjugate beam are 5 V M + EI EI M + M EI EI ppling guiding rules 9 and 1 in the CM, e have V and M. We report that 5M EI M EI ssessment of effetiveness. s before, e see that the method of model formulas enables one to diretl rite the pertinent equations and solve them to obtain the solutions. The onjugate beam method does not require the use of an exerpt of the model formulas. Hoever, the CM requires the appliation of the guiding rules to first onstrut the onjugate beam for the given beam, then rite the equations of equilibrium from the free-bod diagram for the onjugate beam, solve the equations, and appl guiding rules 9 and 1 in the CM to get the slopes and defletions of the beam. oth MoMF and CM ield the same solutions and are equall effetive in solving the problem in this example. Example. ontinuous beam C ith onstant flexural rigidit EI and total length has a roller support at, a roller support at, and a fixed support at C. This beam arries a linearl distributed load and is shon in Fig. 15. Determine (a) the vertial reation fore and slope at, (b) the vertial reation fore and slope at. Fig. 15. Continuous beam C arring a linearl distributed load Solution. The free-bod diagram of the beam is shon in Fig. 16. We readil note that the beam is statiall indeterminate to the seond degree. Page

14 Fig. 16. Free-bod diagram of the ontinuous beam C Using MoMF: In appling the method of model formulas to this beam, e notie that the beam C has a total length, hih ill be the value for the parameter in all model formulas in Eqs. (1) through (). We see that the shear fore V at left end is equal to, the moment M and defletion at are zero, the defletion at is zero, and the slope C and defletion C at C are zero. ppling the model formulas in Eqs. () and () to the beam C and using Eq. () to impose the ondition that ( ) at, in that order, e rite ( ) / ( ) / ( ) EI EI EI ( /) + ( ) + ( ) + 6 EI EI ( ) ( ) / ( ) + ( ) + + ( ) + ( ) ( ) EI 1 EI ( /) 5 + ( ) + ( ) + EI 1 EI ( ) / 5 / ( /) EI 1 EI These three simultaneous equations ield EI 56 Using these values and appling the model formula in Eq. (1), e rite / ( /) EI 6 EI EI x EI We report that 9 1 1EI EI Page

15 Using CM: In using the onjugate beam method to solve the problem in this example, e first make use of the free-bod diagram in Fig. 16 and appl guiding rules 1,,, 5, and 6 in the CM, as listed at the end of Set I, to onstrut the onjugate beam for the given beam C as shon in Fig. 17. eause of the inurred omplexit of the elasti eights dran b parts, e here let the onjugate beam arr the sum of to sets of elasti eight as indiated in Fig. 17. Notie that (a) the simple support at the end in the given beam C in Fig. 15 remains a simple support at the end in the onjugate beam C in Fig. 17; (b) the simple support at in Fig. 15, hih is not at the end of the beam, beomes an unsupported hinge at in Fig. 17; () the fixed end at C in Fig. 15 beomes a free end at C in Fig. 17. The free-bod diagram of the onjugate beam C is shon in Fig. 18, hih ontains three unknons:,, and. Fig. 17. Conjugate beam for given beam C Fig. 18. Free-bod diagram of onjugate beam C Next, referring to Fig. 18 and appling guiding rule 8 in the CM, e rite + Σ, for the entire onjugate beam C in Fig. 18: F Page

16 + + EI EI EI + EI EI 1EI + 1EI EI + Σ, for just segment the left segment of the onjugate beam in Fig. 18: M + + EI EI 5 1EI + Σ, for just segment C the right segment of the onjugate beam in Fig. 18: M + EI EI + EI EI EI 1EI EI The above three simultaneous equations ield EI ppling guiding rules 9 in the CM, e have V. We report that 9 1 1EI 1 56 ppling guiding rules 9 in the CM again, e rite V + EI EI 1EI 168EI We report that 168EI ssessment of effetiveness. One again, e see that the method of model formulas enables one to diretl rite the pertinent equations and solve them to obtain the solutions. In MoMF, the slope and defletion at an position of the beam an alas be evaluated b appling the model formulas in Eqs. (1) and (). The onjugate beam method does not require the use of an exerpt of the model formulas. Hoever, the CM requires the appliation of the guiding rules to first onstrut the onjugate beam for the given beam, then rite the equations of equilibrium from the free-bod diagram for the onjugate beam, solve the equations, and appl guiding rules 9 and 1 in the CM to get the slopes and defletions of the beam. oth MoMF and CM ield the same solutions in this example. eause of the partiular loading on this seond-degree statiall indeterminate beam, the CM has to go an extra mile to onstrut the rather hallenging onjugate beam for the given beam. The riting of moment equilibrium equations for this onjugate beam is also rather hallenging. The required algebrai ork to get the solutions in these to methods is about the same. ll onsidered, one ould likel sa that the overall effort required to obtain the solution is more in using the CM than in using the MoMF in this example. Page

17 III. Conluding Remarks In the method of model formulas, no expliit integration or differentiation is involved in appling an of the model formulas. The model formulas essentiall serve to provide material equations besides the equations of stati equilibrium of the beam that an readil be ritten. Seleted model applied loads are illustrated in Fig. 1(a), hih over most of the loads enountered in undergraduate Mehanis of Materials. Naturall, the MoMF an serve to provide independent heks of solutions obtained using other methods. In the ase of a nonlinearl distributed load on the beam, the model formulas ma be modified b the user for suh a load. Westergaard s onjugate beam method emplos support onditions in the solutions of problems involving defletions of beams. This approah orks ell beause boundar onditions have, in fat, been taken into aount hen the support onditions are speified. The CM usuall requires no expliit integration in the solution, and it requires good skills (a) in draing bending moment diagrams b parts for setting the elasti eights on the onjugate beams, (b) in riting equations of statis equilibrium in the proess of solution. The MoMF and the CM are about equall effetive in solving problems involving statiall indeterminate reations and defletions of beams, exept that the CM is a unique method that an be used to solve defletions of beams in neutral equilibrium. To kno more about this unique feature and apabilit of the CM, refer to the paper b Jong. 1 oth of the MoMF and the CM are suitable for learning b sophomores and juniors; and the have been taught and tested in the ourse Mehanis of Materials at the author s institution for several ears. Referenes 1. H. M. Westergaard, Defletions of eams b the Conjugate eam Method, Journal of the Western Soiet of Engineers, Vol. XXVI, No. 11, pp , S. Timoshenko and G. H. MaCullough, Elements of Strength of Materials (rd Edition), Van Nostrand Compan, In., Ne York, NY, S. H. Crandall, C. D. Norman, and T. J. ardner, n Introdution to the Mehanis of Solids (nd Edition), MGra-Hill, Ne York, NY, R. J. Roark and W. C. Young, Formulas for Stress and Strain (5th Edition), MGra-Hill, Ne York, NY, F.. Singer and. Ptel, Strength of Materials (th Edition), Harper & Ro, Ne York, NY, Ptel and J. Kiusalaas, Mehanis of Materials, rooks/cole, Paifi Grove, C,. 7. J. M. Gere, Mehanis of Materials (6th Edition), rooks/cole, Paifi Grove, C,. 8. F. P. eer, E. R. Johnston, Jr., J. T. DeWolf, and D. F. Mazurek, Mehanis of Materials (5th Edition), MGra- Hill, Ne York, NY, R. G. udnas and J. K. Nisbett, Shigle s Mehanial Engineering Design (8th Edition), MGra-Hill, Ne York, NY, H. T. Grandin, and J. J. Renis, Ne pproah to Solve eam Defletion Problems Using the Method of Segments, Proeedings of the 6 SEE nnual onferene & Exposition, Chiago, I, 6. Page

18 11. I. C. Jong, Defletion of a eam in Neutral Equilibrium à la Conjugate eam Method: Use of Support, Not oundar, Conditions, 7 th SEE Global Colloquium on Engineering Eduation, Cape Ton, South fria, Otober 19-, I. C. Jong, n lternative pproah to Finding eam Reations and Defletions: Method of Model Formulas, International Journal of Engineering Eduation, Vol. 5, No. 1, pp. 65-7, I. C. Jong, Determining Defletions of Elasti eams: What Can the Conjugate eam Method Do That ll Others Cannot? International Journal of Engineering Eduation, Vol. 6, No. 6, pp. 1-17, I. C. Jong, and W. T. Springer, Teahing Defletions of eams: Comparison of dvantages of Method of Model Formulas versus Method of Superposition, Session W55, Proeedings of 11 SEE nnual Conferene & Exposition, Vanouver, C, Canada, June 6-9, 11. Page