AC : DEFLECTION OF A BEAM IN NEUTRAL EQUILIBRIUM À LA CONJUGATE BEAM METHOD: USE OF SUPPORT, NOT BOUNDARY, CONDITIONS

Transcription

1 8-796: DEFETION OF E IN NEUTR EQUIIRIU À ONJUGTE E ETHOD: USE OF SUPPORT, NOT OUNDRY, ONDITIONS Ing-hang Jong, Universit of rkansas Ing-hang Jong serves as Professor of ehanial Engineering at the Universit of rkansas. He reeived a SE in 1961 from the National Taiwan Universit, an SE in 196 from South Dakota Shool of ines and Tehnolog, and a Ph.D. in Theoretial and pplied ehanis in 1965 from Northwestern Universit. He was hair of the ehanis Division, SEE, in His researh interests are in mehanis and engineering eduation. merian Soiet for Engineering Eduation, 8 Page 1.5.1

2 Defletion of a eam in Neutral Equilibrium à la onjugate eam ethod: Use of Support, Not oundar, onditions bstrat eams with flexural rigidit will deflet under loading. Is it possible to asertain the defletion of a loaded beam in neutral equilibrium? The answer is es aording to the onjugate beam method, but a resounding no aording to all other established methods. The objetive of this paper is to share with fellow engineering eduators the insights, highlights, and several illustrative examples for teahing the onjugate beam method. In partiular, it is pointed out that (a) support onditions (or tpes), rather than boundar onditions, are what the onjugate beam method needs in finding solutions for defletions of loaded beams, (b) more support onditions than boundar onditions are usuall known for beams in neutral equilibrium, and () the onjugate beam method often works better than other established methods in determining defletions of beams. It is demonstrated in this paper that the onjugate beam method does find the likel, or unique, defletion of a loaded beam in neutral equilibrium. I. Introdution ll beams onsidered in this paper are elasti beams, whih are longitudinal members subjeted to transverse loads and are usuall in stati equilibrium. 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. Fig. 1 tual beam Fig. onjugate beam The beam in Fig. 1 is in neutral equilibrium and will adopt a defleted shape. Is it possible to asertain the defletion of a loaded beam in neutral equilibrium? The answer is es aording to the onjugate beam method, 1 4 but a resounding no aording to all other established methods, 1 suh as (a) method of double integration (with or without the use of singularit funtions), (b) method of superposition, () method using moment-area theorems, (d) method using astigliano s theorem, and (e) method of segments. These other methods all expet a beam to have suffiient well-defined boundar onditions for use in seeking a unique solution for the defletion of the beam. The beam in Fig. 1 manifests onl one known boundar ondition (i.e., the defletion at the hinge support is zero), whih is simpl insuffiient to allow the other methods to settle on a unique solution. However, the onjugate beam method has no trouble with the beam in Fig. 1. This beam manifests three support onditions (i.e., free end at, simple support at, and free end at ), whih are suffiient to allow a orresponding onjugate beam to be onstruted as shown in Fig.. For now, the atual beam and the onjugate beam in Figs. 1 and are used to Page 1.5.

3 serve introdutor purposes. Finding of the defletion of this beam in neutral equilibrium is deferred to Example 5 later. The onjugate beam method was first propounded in 191 b Westergaard. 1 One an find brief presentations of this method in earlier mehanis of materials textbooks b Timoshenko and aullough and b Singer and Ptel 4. Reentl, a set of ten guiding rules to failitate the use of this method was snthesized b Jong from the original paper of Westergaard. 1 For benefit of a wider readership with different speialties, a brief summar of the guiding rules needed in this method is inluded. Readers, who are familiar with the rudiments of this method, ma skip the part presented in Setion II. This paper is intended to share with fellow engineering eduators the insights, highlights, and several illustrative examples for teahing students that (a) the onjugate beam method is a natural and logial extension of the method using moment-area theorems, (b) b observing the support onditions of an atual beam (as in Fig. 1) and appling the guiding rules in this method, the onjugate beam for an atual beam an readil be onstruted (as in Fig. ), () slopes and defletions of an atual beam are simpl obtained from the shearing fores and bending moments in the orresponding positions of the onjugate beam, (d) support onditions, rather than boundar onditions, are what the onjugate beam method needs in generating solutions for slopes and defletions of atual beams, (e) more support onditions than boundar onditions are usuall known for beams in neutral equilibrium, and (f) the onjugate beam method an do whatever other established methods an do, and more, in determining defletions of beams. The paper demonstrates that the onjugate beam method does find the unique defletion of a loaded beam in neutral equilibrium. The results obtained are assessed analtiall b omparison with well-known results in textbooks. II. Guiding Rules in onjugate eam ethod lthough Westergaard 1 propounded the onjugate beam method in a 8-page paper, earlier textbooks,4 provided mainl brief and elementar presentations of this method. Without adequate guiding rules and a good number of tpial examples, the onjugate beam method ma likel appear as inaessible or esoteri to man beginners. On the other hand, most beginners are pleasantl surprised to learn that there are onl two major steps in this method. The first step is to set up an additional beam, alled onjugate beam, besides the atual beam. The seond step is to determine the shearing fores and bending moments in the onjugate beam using mainl onepts and skills in statis. In the proess, these two steps are most effetivel guided b the set of ten rules snthesized b Jong. These rules are natural and logial extensions of the method using moment-area theorems. 11 The are summarized as follows: Rule 1: The onjugate beam and the atual beam are of the same length. Rule : The loading on the onjugate beam is simpl the distributed elasti weight, whih is given b the bending moment in the atual beam divided b the flexural rigidit EI of the atual beam. (The elasti weight, /EI, points upward if the bending moment is positive to ause the top fiber in ompression in beam onvention.) For eah existing support ondition of the atual beam, there is a orresponding support ondition for the onjugate beam. The orrespondene is given b rules through 7 listed in Table 1, Page 1.5.

4 where a simple support is either a roller support or a hinge support, sine a beam is usuall not subjeted to axial loads. Table 1 orresponding support ondition for the onjugate beam Existing support ondition of the atual beam Rule : Fixed end Rule 4: Free end Rule 5: Simple support at the end Rule 6: Simple support not at the end Rule 7: Unsupported hinge orresponding support ondition for the onjugate beam Free end Fixed end Simple support at the end Unsupported hinge Simple support The slope and defletion of the atual beam are obtained b emploing the following rules: Rule 8: The onjugate beam (hene its free bod) is in stati equilibrium. Rule 9: The slope of (the enterline of ) the atual beam at an ross setion is given b the shearing fore at that ross setion of the onjugate beam. (This slope is positive, or ounterlokwise, if the shearing fore is positive tending to rotate the beam element lokwise in beam onvention.) Rule 1: The defletion of (the enterline of ) the atual beam at an point is given b the bending moment at that point of the onjugate beam. (This defletion is upward if the bending moment is positive tending to ause the top fiber in ompression in beam onvention.) III. ppliations of onjugate eam ethod For better understanding of the method, this setion inludes several detailed examples with different degrees of omplexit and hallenge. In appliations of the onjugate beam method, more statis skills and ver little expliit alulus skills are usuall needed. With the guiding rules, as summarized in Setion II, one ma here see that the onjugate beam method is aessible and eas to appl to find defletions of beams. Example 1. antilever beam with total length and onstant flexural rigidit EI arries a onentrated fore P at as shown in Fig.. Determine (a) the slope and defletion at, (b) the slope and defletion at. Fig. antilever beam (atual beam) Solution. In aordane with rules 1 through 4 in Setion II, we first onstrut in Fig. 4 the onjugate beam (i.e., the additional beam) for the atual beam in Fig.. Notie that the onjugate beam has the same length as the atual beam, and it arries a linearl distributed elasti weight pointing downward with intensit equal to zero at and equal to /EI at. Page 1.5.4

5 Fig. 4 onjugate beam (additional beam) for the beam in Fig. Furthermore, notie that the atual beam in Fig. has a free end at and a fixed end at, while the onjugate beam in Fig. 4 has a fixed end at and a free end at. Next, we draw in Fig. 5 the free-bod diagram for the onjugate beam, where the supersript is used to signif a quantit assoiated with the onjugate beam. Suh a notation is needed for distinguishing a quantit assoiated with the onjugate beam from the fore or moment ating on the atual beam. Fig. 5 Free-bod diagram for the onjugate beam in Fig. 4 rule 8 in Setion II, the onjugate beam (hene its free bod) is in stati equilibrium. Referring to Fig. 5, we write F : EI : EI EI 5 6EI EI 5 6EI rules 9 and 1 in Setion II, the slope and the defletion at the free end of the atual beam in Fig. are, respetivel, given b the shearing fore V and the bending moment at the fixed end of the onjugate beam in Fig. 4. We write We report that V EI EI 5 6EI 5 6EI Note that the slope is ounterlokwise beause the shearing fore tends to rotate the beam element at lokwise and is positive, while the defletion is downward beause the bending moment tends to ause tension in the top fiber of the beam at and is negative. ppling rules 9 and 1 in Setion II and referring to Fig. 5, we determine the slope and defletion at as follows: We report that V EI 5 6EI EI EI Page 1.5.5

6 EI EI These results are illustrated in Fig. 6. Fig. 6 Defletion of the antilever beam in Fig. Remark: The preeding results for the antilever beam in Fig. are in agreement with slopes and defletions for antilever beams ontained in a table or an appendix of textbooks. 4,11 Example. simpl supported beam with length and onstant flexural rigidit EI arries a onentrated fore P at as shown in Fig. 7. Determine (a) the slopes,, at,, ; (b) the defletion at. Fig. 7 simpl supported beam (atual beam) Solution. The reations at the simple supports and of this beam an readil be determined from statis equilibrium of the beam and are shown in the free-bod diagram in Fig. 8. Fig. 8 Free-bod diagram for the beam in Fig. 7 ppling rules 1,, and 5 in Setion II, we readil onstrut in Fig. 9 the onjugate beam for the atual beam in Fig. 7, where the simple supports at the ends remain unhanged and the loading diagram showing the distributed elasti weight on the onjugate beam is drawn b parts. Fig. 9 onjugate beam (additional beam) for the beam in Fig. 7 Page 1.5.6

7 Fig. 1 Free-bod diagram for the onjugate beam in Fig. 9 rule 8 in Setion II, the onjugate beam (hene its free bod) is in stati equilibrium. Referring to Fig. 1, we write : : F 1 1 EI 9 EI 1 1 EI EI 5 81EI 4 81EI rule 9 in Setion II, slopes and of the atual beam at and are given b shearing fores V and V in the onjugate beam at and, respetivel. We write 4 V 81EI V 5 81EI ppling rules 9 and 1 in Setion II and referring to Fig. 1, we determine the slope and defletion at as follows: 1 V EI 81EI EI 4EI We report that 4 81EI 5 81EI 81EI 4 4EI These results are illustrated in Fig. 11. Fig. 11 Defletion of the simpl supported beam in Fig. 7 Remark: The preeding results are in agreement with slopes and defletions for simple beams ontained in a table or an appendix of textbooks. 4,11 Page 1.5.7

8 Example. Gerber beam (Gerberbalken) with total length 4 has a hinge onnetion at and onstant flexural rigidit EI in its segments and DE. This beam is supported and loaded with a onentrated moment at D as shown in Fig. 1. Determine (a) the slopes, D, and E at, D, and E, respetivel; (b) the slope ( ) ljust to the left of ; () the slope ( ) r just to the right of ; (d) the defletion at ; (e) the defletion D at D. Fig. 1 Gerber beam (atual beam) Solution. Sine the bending moment at the hinge must be zero, we readil find that the reation fore at the right end E is E /( ). learl, this beam is statiall indeterminate to the first degree. We note in Fig. 1 that the support onditions in the atual beam are as follows: a fixed support at, a simple support (not at the end) at, an unsupported hinge at, a simple support at the end E. In aordane with rules 1,,, 5, 6, and 7 in Setion II, we draw in Fig. 1 the onjugate beam for the atual beam in Fig. 1. Notie in Fig. 1 that the loading diagram showing the distributed elasti weight on the onjugate beam is drawn b parts. Fig. 1 onjugate beam (additional beam) for the Gerber beam in Fig. 1 Furthermore, notie in Fig. 1 that, b rules, 5, 6, and 7 in Setion II, the support onditions in the onjugate beam are as follows: a free end at, an unsupported hinge at, a simple support at, a simple support at the end E. To learl show how this seemingl omplex problem is solved, we next draw in Fig. 14 the free-bod diagram for the onjugate beam, where the unknowns are,, and E. Page 1.5.8

9 Fig. 14 Free-bod diagram for the onjugate beam in Fig. 1 Note in Fig. 14 that the bending moment at the hinge must be zero. rule 8 in Setion II, the onjugate beam (hene its free bod) is in stati equilibrium. These onditions allow us to write the equations and solutions for the unknowns,, and E as follows: : EI EI EI 4EI 5 E : EI EI EI 16EI 4 : F EI EI EI 4 E E 16EI ppling rule 9 in Setion II and referring to Fig. 14, we write 5 V EI EI EI EI 4 EI 5 16EI D V D E D E VE E E 16EI 8EI We report that ( ) l ( V ) l EI EI EI EI ( ) ( V ) E EI EI r r r ( ) l 8EI ( ) 16EI Page 1.5.9

10 8EI D 5 E 16EI 16EI ( ) 8EI ( ) 16EI l r ppling rule 1 in Setion II and referring to Fig. 14, we determine the defletions at and at D as follows: D EI EI E D D E EI We report that 7 4EI D D 7 48EI 7 4EI 7 48EI ased on these results, we an plot in Fig. 15 the defletion of the Gerber beam in Fig. 1. Fig. 15 Defletion of the Gerber beam in Fig. 1 Remark: Nine of the ten guiding rules have been used in this example. The results have been verified b the author to be in agreement with results obtained using moment-area theorems. 11 Example 4. fix-ended beam with length and onstant flexural rigidit EI is ated on b a onentrated moment at its midpoint, and its right fixed end is fored to shift upward b an amount, without rotation, as shown in Fig. 16. Determine (a) the vertial reation fore and the reation moment at, (b) the vertial reation fore and the reation moment at. Fig. 16 Relative vertial shifting of supports in a loaded beam (atual beam) Solution. The free-bod diagram for this beam ma be drawn as shown in Fig. 17, where the unknowns are,,, and. learl, this beam is statiall indeterminate to the seond degree. Page 1.5.1

11 Fig. 17 Free-bod diagram for the beam in Fig. 16 For equilibrium of the atual beam, we refer to Fig. 17 to write : (a) F : (b) The beam in Fig. 16 has fixed ends at and, but the fixed end is fored to move upward, without rotation, an amount of. This means that the onjugate beam for the atual beam must, b rules 1,,, and 1 in Setion II, have free ends at and plus a ounterlokwise bending moment of magnitude ating at, as shown in Fig. 18, where the distributed elasti weight on the onjugate beam is drawn b parts. Fig. 18 onjugate beam (additional beam) for the beam in Fig. 16 The reason we appl rule 1 in Setion II to impose a ounterlokwise bending moment of magnitude ating at in Fig. 18 is to take into aount the extraordinar boundar ondition of upward displaement of the fixed end, without rotation, of the atual beam. [If an atual beam should have a speified slope (or rotation) at an point, rule 9 in Setion II would, of ourse, be applied to impose a shearing fore at that point of the onjugate beam.] rule 8 in Setion II, the free bod of the onjugate beam in Fig. 18 is in stati equilibrium without an support in spae. We refer to this figure to get the two needed additional equations: : F : () EI EI EI (d) EI EI 4 EI Solving the preeding Eqs. (a) through (d) simultaneousl, we obtain 4EI 4EI 4 From these results and the assumed diretions of fores and moments in Fig. 17, we report that Page

12 4EI 4EI 4 4EI 4EI 4 Remark: Equations () and (d) in this example an be obtained b imposing the onditions / (for the relative angle between the two tangents drawn at points and of the beam) and t / (for the tangential deviation of point with respet to the tangent drawn at point of the beam), respetivel, in the method using moment-area theorems. 11 learl, the same results in this example are what will be obtained in the method using moment-area theorems. Example 5. beam with total length and onstant flexural rigidit EI is supported on a single simple support at its midpoint as shown in Fig. 19. The beam is in neutral equilibrium beause it arries a onentrated moment at and a onentrated fore P at, where P /. Determine (a) the slope and defletion at, (b) the slope and defletion at, () the slope at. Fig. 19 tual beam in neutral equilibrium (repeat of Fig. 1) Solution. This example emplos the beam in Fig. 1, whih was used earlier for introdutor purposes. The problem in this example annot be takled b an method other than the onjugate beam method. This beam manifests three support onditions (i.e., free end at, simple support at, and free end at ), whih are suffiient to allow a orresponding onjugate beam to be onstruted as shown in Fig., whih was shown as Fig. earlier for introdutor purposes. Fig. onjugate beam for the beam in Fig. 19 (repeat of Fig. ) Fig. 1 Free-bod diagram for the onjugate beam in Fig. Page 1.5.1

13 To learl show how this seemingl baffling problem is solved, we draw in Fig. 1 the free-bod diagram for the onjugate beam, where the unknowns are,,, and. rule 8 in Setion II, the free bod in Fig. 1 is in stati equilibrium. oreover, the bending moment at the hinge in Fig. 1 must be zero. These onditions allow us to write F, for the entire onjugate beam in Fig. 1: EI EI (a), for just segment the left segment of the onjugate beam in Fig. 1: (b) EI, for just segment the right segment of the onjugate beam in Fig. 1: () EI The above three equations ontain four unknowns:,,, and. Thus, we are faed with a problem involving a onjugate beam that is statiall indeterminate to the first degree. The statial indetermina of the onjugate beam in Fig. an, of ourse, be resolved b using an of the established methods. et us hoose to emplo the onjugate beam method further to generate the needed additional equation to go with the preeding three equations (a), (b), and () for solving the problem in this example. For simpliit, the flexural rigidit of eah segment of the onjugate beam in Fig. ma be taken as equal to 1 unit. drawing the elasti weight b parts, we onstrut in Fig. the onjugate beam for the onjugate beam in Fig.. Note that suh a onjugate beam as shown in Fig. has free ends at and and a single simple support at its midpoint. Fig. onjugate beam for the onjugate beam in Fig. rule 8 in Setion II, the onjugate beam in Fig. must be in stati equilibrium. Similar to the original atual beam in Fig. 19, the onjugate beam in Fig. turns out to be also in neutral equilibrium. Using the supersripts to refer to the onjugate beam for the onjugate beam and referring to Fig., we write : Page 1.5.1

14 4 6EI 5 4 6EI Note that Eq. (d) is the additional equation needed to go with the preeding Eqs. (a), (b), and () to resolve the statial indetermina mentioned above. (d) Fig. Defletion of the onjugate beam in Fig. The raison d être for the above obtained Eq. (d) ma briefl be examined. When the onjugate beam under elasti weight in Fig. deflets, it will adopt a shape as illustrated in Fig.. Those familiar with the method using moment-area theorems 11 will readil pereive that the tangential deviation t / deviation t / of point with respet to the tangent drawn at point is equal to the tangential of point with respet to the tangent drawn at point ; i.e., t t / / ording to the seond moment-area theorem, 11 the tangential deviation t / is equal to the first moment, taken ounterlokwise about point, of the elasti weight between points and in Fig.. eantime, the tangential deviation t / is equal to the first moment, taken lokwise about point, of the elasti weight between points and in Fig.. transposing terms in the above equation, we see that t t / / arring out the seond moment-area theorem for terms in this equation, we see that this equation leads to the same equation as Eq. (d) above. Thus, appling the onjugate beam method to further stud the statiall indeterminate onjugate beam in Fig. is sound and well. Solving the preeding Eqs. (a) through (d) simultaneousl for the four unknowns in them, we get 19 EI 11 EI 9 EI 6EI Using these results and appling rules 9 and 1 in Setion II, we an refer to Fig. 1 and write 19 EI 11 EI V V 9 7 EI 6EI V EI EI 6EI Page

15 For the loaded beam in neutral equilibrium in Figs. 1 and 19, we report that 19 EI 11 EI EI 7 6EI 6EI ased on these results, we depit in Fig. 4 the defletion of the atual beam in Figs. 1 and 19. Fig. 4 Defletion of the atual beam in Figs. 1 and 19 Remark: Sine the problem in this example annot be solved b an other methods, no diret omparison of the preeding results an be made. Nonetheless, assessment of these results is possible as presented in Setion IV. IV. ssessment of Results Obtained in Example 5 et us refer to both Fig. 19 and Fig. 4. Sine we have obtained the slope for the tangent drawn at in Fig. 4, we ma perform an analtial hek of the solutions b regarding the defleted shape of this beam as the elasti urve of two antilever beams: a antilever beam with length, fixed at, whih is defleted from to b a onentrated moment at ; a antilever beam with length, fixed at, whih is defleted from to b a onentrated fore P at, where P /. For ease of referene, we repeat the beam and the results as shown below. Fig. 19 tual beam in neutral equilibrium (repeat of Fig. 1) 19 EI 11 EI EI 7 6EI From the geometr in Fig. 4 and the results as shown, we find the following: 6EI Page

16 EI 7 EI 6 EI EI 19 / EI EI EI ( ) 6EI EI EI EI EI We have 11 ( ) EI EI EI EI EI / EI / EI EI / EI We note that the above values for, /,, and / are all in agreement with those found in a table or an appendix of textbooks 4,11 for the defletion and slope of the free end of a antilever beam loaded at its free end with (a) a onentrated moment, (b) a onentrated fore P, respetivel. Note that no rigorous experimental results for defletions of beams in neutral equilibrium are readil available. In the absene of an available results for diret omparison, the foregoing agreeable assessment ma be taken as a pat on the bak for the efforts and results obtained in solving a baffling problem in Example 5. V. onluding Remarks This paper is written to share with fellow engineering eduators the insights, highlights, and five detailed illustrative examples for teahing the onjugate beam method. It is pointed out that more support onditions than boundar onditions are usuall known for beams in neutral equilibrium. The onjugate beam method an readil handle the following five basi support onditions (or tpes): (i) fixed end, (ii) free end, (iii) simple support at the end, (iv) simple support not at the end, and (v) unsupported hinge. Furthermore, rules 9 and 1 in Setion II an readil be applied to address extraordinar boundar onditions, where speifi values for slopes and defletions are present or stipulated in the problem. This is a versatile feature inherent in the guiding rules for the onjugate beam method, as illustrated in Example 4. esides being able to orretl find solutions for omplex as well as simple problems of defletions of beams, as illustrated in Examples 1 through 4, the onjugate beam method stands out as the onl method that is able to pursue and ield the solution for the defletion of a loaded beam in neutral equilibrium, as illustrated in Example 5. This method is unique and outstanding. The Page

17 root ause ontributing to this rather unusual senario lies in the use of support onditions in the onjugate beam method versus the use of boundar onditions in all other methods. In using the onjugate beam method, more statis skills and ver little expliit alulus skills are usuall needed. When the support onditions of a beam are properl reognized and taken into aount b using the guiding rules in the onjugate beam method, the boundar onditions will, of ourse, be satisfied automatiall! In other words, all solutions generated b the onjugate beam method have satisfied the boundar onditions in the beginning stages of the solutions. The onjugate beam method has been demonstrated in this paper to do a job as good as or better than other established methods in determining defletions of beams. For ears, the author has taught the onjugate beam method, in addition to the established methods using double integration and moment-area theorems, in his own lass of mehanis of materials at his home institution, where the students have been provided with the 1 Guiding Rules in onjugate eam ethod on a single page. Overwhelmingl, his students favor the onjugate beam method over the other methods in determining defletions of beams. Referenes 1. H.. Westergaard, Defletions of eams b the onjugate eam ethod, Journal of the Western Soiet of Engineers, Vol. XXVI, No. 11, pp , I.. Jong, Effetive Teahing and earning of the onjugate eam ethod: Snthesized Guiding Rules, Proeedings of the 4 SEE nnual onferene & Exposition, Salt ake it, UT, 4.. S. Timoshenko and G. H. aullough, Elements of Strength of aterials (rd Edition), Van Nostrand, New York, NY, F.. Singer and. Ptel, Strength of aterials (4th Edition), Harper & Row, New York, NY, S. H. randall,. D. Norman, and T. J. ardner, n Introdution to the ehanis of Solids (nd Edition), Graw-Hill, New York, NY, R. J. Roark and W.. Young, Formulas for Stress and Strain (5th Edition), Graw-Hill, New York, NY, Ptel and J. Kiusalaas, ehanis of aterials, rooks/ole, Paifi Grove,,. 8. J.. Gere, ehanis of aterials (6th Edition), rooks/ole, Paifi Grove,, H. T. Grandin and J. J. Renis, New pproah to Solve eam Defletion Problems using the ethod of Segments, Proeedings of the 6 SEE nnual onferene & Exposition, hiago, I, I.. Jong, J. J. Renis, and H. T. Grandin, Jr., New pproah to nalzing Reations and Defletions of eams: Formulation and Examples, Proeedings of IEE6, SE International ehanial Engineering ongress and Exposition, hiago, I, F. P. eer, E. R. Johnston, Jr., and J. T. DeWolf, ehanis of aterials (4th Edition), Graw-Hill, New York, NY, R. G. udnas and J. K. Nisbett, Shigle s ehanial Engineering Design (8th Edition), Graw-Hill, New York, NY, 8. Page