Teaching Deflections of Beams: Advantages of Method of Model Formulas versus Method of Integration

Transcription

1 Teaching Deflections of eams: dvantages of Method of Model Formulas versus Method of Integration Ing-hang Jong, William T. Springer, Rick J. ouvillion Universit of rkansas, Faetteville, R 77 bstract The method of model formulas is a ne method for solving staticall indeterminate reactions and deflections of elastic beams. Since its publication in a recent issue of the IJEE, man instructors of Mechanics of Materials have considerable interest in knoing an effective a for teaching this method to enrich students stud and their set of skills in determining beam reactions and deflections. Moreover, people are interested in seeing demonstrations shoing an advantage of this method over the traditional methods. This paper is aimed at (a providing comparisons of this ne method versus the traditional method of integration via several head-to-head contrasting solutions of same problems, and (b proposing a set of steps for use to effectivel introduce and teach this ne method to students. It is a considered opinion that the method of model formulas be taught to students after having taught them one or more of the traditional methods. 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. Traditional methods used in determining staticall indeterminate reactions and deflections of elastic beams include: - method of integration (ith or ithout use of singularit functions, method of superposition, method using moment-area theorems, method of conjugate beam, method using astigliano s theorem, and method of segments. The method of model formulas is a nel propounded method. eginning ith an elastic beam under a preset general loading, a set of four model formulas are derived and established for use in this ne method. These 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 kno 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., courtes of IJEE. roceedings of the Midest Section onference of the merican Societ for Engineering Education

2 Excerpt from the Method of Model Formulas ourtes: Int. J. Engng. Ed., Vol. 5, No., pp. 65-7, 9 (a (b ositive directions of forces, moments, slopes, and deflections Va M a a x x xx K xx x x K EI EI EI EI xx xu xu EI x EI x m m xxm xu m EI EI ( u ( u ( V M K x x x xx xx x x EI EI EI x x xu xu EI ( m m x xm xum a a a a K 5 5 ( u x EI EI u x ( Va L Ma L ( L x K ( L xk ( L x b a EI EI EI EI 6 EI ( L x ( Lu EI ( Lu m m ( Lxm ( Lum EI EI ( ux EI ( ux ( VL a MaL K b a al ( Lx ( Lx ( Lx EI EI EI 5 5 ( L x ( Lu ( Lu EI( u x EI EI( u x m m ( Lxm ( Lum K ( Fig.. Loading, deflections, and formulas in the Method of Model Formulas for beams roceedings of the Midest Section onference of the merican Societ for Engineering Education

3 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: 8,9 n n xa ( xa if xa and n (5 n xa if xa and n (6 n xa if xa or n (7 x n n x a dx x a n if n (8 x n n xa dx xa if n (9 d n n xa n xa dx if n ( d n n xa xa dx if n ( Equations (6 and (7 impl that, in using singularit functions for beams, e take b for b ( b for b ( 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.. 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. (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. (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. (a]. concentrated force or a distributed force applied to the beam is positive if it is directed donard [e.g., the concentrated force, the linearl distributed force ith intensit on the left side and intensit on the right side in Fig. (a, here the distribution becomes uniform if ]. The slopes and deflections of a beam displaced from to ab are shon in Fig. (b. Note that positive slope is a counterclockise angular displacement [e.g., a and b in Fig. (b]. positive deflection is an upard linear displacement [e.g., a and b in Fig. (b]. roceedings of the Midest Section onference of the merican Societ for Engineering Education

4 II. Teaching and Learning a Ne Method via ontrast beteen Solutions Equations ( through ( are related to the beam and loading shon in Fig. ; the are the model formulas in the ne method. Their derivation (not a main concern in this paper can be found in the paper that propounded the method of model formulas. Note that L in the model formulas in Eqs. ( through ( is a parameter representing the total length of the beam. In other ords, L is to be replaced b the total length of the beam segment, to hich the model formulas are applied. Staticall indeterminate reactions as ell as slopes and deflections of beams can, of course, be solved. 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. Having learned an additional efficacious method, students stud and set of skills are enriched. Mechanics is mostl a deductive science, but learning is mostl an inductive process. For the purposes of teaching and learning, all examples ill be first solved b the traditional method of integration (MoI ith use of singularit functions then solved again b the method of model formulas (MoMF. s usual, the loading function, shear force, bending moment, slope, and deflection of the beam are denoted b the smbols q, V, M,, and, respectivel. Example. simpl supported beam D ith constant flexural rigidit EI and length L is acted on b a concentrated force at and a concentrated moment at 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 in static equilibrium. Its free-bod diagram is shon in Fig.. Fig.. Free-bod diagram of the simpl supported beam D Using MoI: Using the smbols defined earlier and appling the method of integration (ith use of singularit functions to this beam, e rite 5 L L q x x x 5 L L V x x x 5 L x L M EI x x roceedings of the Midest Section onference of the merican Societ for Engineering Education

5 5 5 L L 6 5 L L 8 6 EI x x x EI x x x x The boundar conditions of this beam reveal that ( at and ( L at D. Imposing these to conditions, respectivel, e rite 5 L L ( 8 6 These to simultaneous equations ield 8 L Using these values and the foregoing equations for EI and EI, e rite EI x 8EI 5 L L 7 D ( L xl EI EI EI 5 L L xl/ 8EI EI 8 8EI 7 D 8 Using MoMF: In appling the method of model formulas to this beam, e must adhere to the sign conventions as illustrated in Fig.. t the left end, the moment M is, the shear force V is 5/, 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 x L / for the concentrated force at and xk L / for the concentrated moment at. ppling the model formulas in Eqs. ( and (, successivel, to this beam D, e rite 5 / ( L L L D L L EI EI EI 5 / ( L L L L L L EI These to simultaneous equations ield 7 D 8EI Using the value of and appling the model formula in Eq. (, e rite L 5 / L xl/ 8 roceedings of the Midest Section onference of the merican Societ for Engineering Education

6 6 8EI 7 D 8 Remark. We observe that both the method of integration (ith use of singularit functions and the method of model formulas ield the same solutions, as expected. In fact, the solution b the MoMF looks more direct than that b the MoI. Furthermore, if singularit functions ere not used in the MoI, the solution ould require division of the beam into multiple segments (such as,, and D, and much more effort in algebraic ork in the solution ould be involved. In Examples through 5, readers ma observe similar features. Example. cantilever beam ith constant flexural rigidit EI and length L is loaded ith a distributed load of intensit in segment as shon in Fig.. Determine (a the slope and deflection at, (b the slope and deflection at. Fig.. antilever beam loaded ith a distributed load Solution. The beam is in static equilibrium. Its free-bod diagram is shon in Fig. 5. Fig. 5. Free-bod diagram of the cantilever beam Using MoI: ppling the method of integration to this beam, e rite qx x L V x x L L M EI x x L EI x x 6 6 L EI x x x The boundar conditions of this beam reveal that ( L and ( L at. Imposing these to conditions, respectivel, e rite L L 6 6 roceedings of the Midest Section onference of the merican Societ for Engineering Education

7 7 These to simultaneous equations ield 7L 8 L L L L 8 Using these values and the foregoing equations for EI and EI, e rite EI x 7L 8 EI x EI L L x L/ EI 8EI L 8EI 7L 8EI x L/ 7 L L L EI EI EI 9EI L 8EI L 8EI 7L 9EI Using MoMF: Let the method of model formulas be no applied to this beam. The shear force V and bending moment M at the free end, as ell as the slope and deflection at the fixed end, are all zero. Noting that 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 7L 8EI EI EI L 8EI Using these values and appling the model formulas in Eqs. ( and (, respectivel, e rite L L xl/ 8EI L L 7L x L/ EI 9EI 7L 8EI L 8EI L 8EI 7L 9EI roceedings of the Midest Section onference of the merican Societ for Engineering Education

8 8 Example. cantilever beam ith constant flexural rigidit EI and total length L is propped at and carries a concentrated moment M at as shon in Fig. 6. Determine (a the vertical reaction force and slope at, (b the slope and deflection at. Fig. 6. antilever beam propped at and carring a moment at Solution. The beam is in static equilibrium. Its free-bod diagram is shon in Fig. 7, here e note that the beam is staticall indeterminate to the first degree. Fig. 7. Free-bod diagram of the propped cantilever beam Using MoI: ppling the method of integration to this beam, e rite q x M xl V x M xl M EI x M x L EI x M x L M EI x xl x 6 The boundar conditions of this beam reveal that ( at, ( L at, and ( L at. Imposing these three conditions, respectivel, e rite ( L ML M ( L L ( L 6 These three simultaneous equations ield M L 9M 8 6L Using these values and the foregoing equations for EI and EI, e rite M L 5ML x L EI 8EI xl EI EI EI ML L L xl EI EI roceedings of the Midest Section onference of the merican Societ for Engineering Education

9 9 9M 6L ML 8 EI 5M L EI ML EI Using MoMF: Let the method of model formulas be no applied to this beam. We note that this beam has a total length of L, hich ill be the value for the parameter L in all the model formulas in Eqs. ( through (. We see that the deflection and the slope at, as ell as the deflection at, are equal to zero. ppling the model formulas in Eqs. ( and ( to the entire beam, e rite ( L M ( L L EI EI ( L EI M ( L ( LL These to simultaneous equations ield 9M 6L M L 8EI Using these values and appling the model formulas in Eqs. ( and (, respectivel, e rite 5ML L x L EI EI ML L L x L EI 9M 6L ML 8 EI 5M L EI ML EI Example. continuous beam ith constant flexural rigidit EI and total length L has a roller support at, a roller support at, a fixed support at and carries a linearl distributed load as shon in Fig. 8. Determine (a the vertical reaction force and slope at, (b the vertical reaction force and slope at. Fig. 8. ontinuous beam carring a linearl distributed load Solution. The beam is in static equilibrium. Its free-bod diagram is shon in Fig. 9, here e note that the beam is staticall indeterminate to the second degree. roceedings of the Midest Section onference of the merican Societ for Engineering Education

10 Fig. 9. Free-bod diagram of the continuous beam Using MoI: Treating as an applied unknon concentrated force, e ma use superposition technique to first rite the loading function q as follos: q x xl x x xl xl L L ppling the method of integration to this beam, e rite V x xl x x xl xl L L M EI x xl x x xl xl L L EI x x L x x x L x L 8L 8L EI x x L x x x L x L x L L The boundar conditions of this beam reveal that ( L and ( L at, ( L at, and ( at. Imposing these four conditions, in order, e rite ( L L ( L ( L L L 8L 8L L L 5 5 ( L L ( L ( L L L ( L 6 8 L The above four simultaneous equations ield 9L 5 L L L L L 56 Using these values and the foregoing equation for EI, e rite L EI x L EI L L L L xl EI 8L 68EI 9L L EI L 56 L 68EI roceedings of the Midest Section onference of the merican Societ for Engineering Education

11 Using MoMF: Let the method of model formulas be no applied to this beam. We notice that the beam has a total length L, hich ill be the value for the parameter L in all the model formulas in Eqs. ( 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 and deflection at are zero. ppling the model formulas in Eqs. ( and ( to the beam and using Eq. ( to impose the condition that ( L at, in that order, e rite L ( / ( / L L L ( L EI EI EIL ( / ( LL ( LL EIL ( ( L / ( ( L ( LL ( L ( L EI EIL ( / 5 ( LL ( LL EI EIL ( / 5 / ( / 6 EI EI EIL 5 L L L L These three simultaneous equations ield 9 L L L EI 56 Using these values and appling the model formula in Eq. (, e rite / ( / L EI EIL 68EI xl L L L 9L L EI L 56 L 68EI Example 5. stepped beam D, propped at and fixed at D, carries a concentrated force at as shon in Fig., here the segments and D have flexural rigidities EI and EI, respectivel. Determine (a the reaction force at ; (b the slopes,, and at,, and ; (c the deflections and at and. Fig.. Stepped beam D being supported at and D and loaded at Solution. The beam is in static equilibrium and is staticall indeterminate to the first degree. To facilitate the analsis of this beam, e first dra the free-bod diagrams of its segments and D as shon in Fig.. roceedings of the Midest Section onference of the merican Societ for Engineering Education

12 (a (b Fig. Free-bod diagrams of the segments and D Using MoI: ppling the method of integration to the segment as shon in Fig. (a, e rite q x xl V x x L M EI x x L EI x x L EI x x L x 6 6 ppling the method of integration to the segment D as shon in Fig. (b, e rite q V xl M xl D V V xl M xl D MD EI D V x L M x L V EI D x L M x L V M EI D xl xl x 6 The boundar conditions of the beam reveal that ( at ; ( L D ( L and ( L D ( L at ; D ( L and D ( L at D. Imposing these five conditions, in order, e rite (a I 6 6 ( L ( L I I ( L ( L ( L ( L I V M (d 6 ( L ( L ( L V (e ( L M ( L For equilibrium of the segment in Fig. (a, e rite (b (c roceedings of the Midest Section onference of the merican Societ for Engineering Education

13 F : V (f M : L LM (g The above seven simultaneous Eqs. (a through (g ield ( I 5 I 9 ( I 8I ( 5 II V 9 ( I 8I M ( I I 9 I 8 I ( I II I I ( 9I 8I ( II 9 ( I 8I ( 89 I 69 ( I 8I Using these values and the foregoing equations for EI and EI e rite ( I II I x EI EI I ( 9I 8I L ( I II I xl EI EI EI I ( 9I 8I 8 ( L ( I I xl EI EI EI EI ( 9I 8I L ( I I I I xl EI EI I ( 9I 8I L 7 7 L L ( I I EI EI ( 9I 8I xl ( I 5 I 9 ( I 8I ( I I I I EI I ( 9I 8I ( I 8 I I I EI I ( 9I 8I ( I I EI ( 9I 8I ( I 7I I 7I EI I ( 9 I 8 I ( I I ( 9I 8I Using MoMF: Let the method of model formulas be no applied to this stepped beam. We first divide the beam into to segments, hose free-bod diagrams are shon in parts (a and (b of Fig.. In particular, note that the segment has a total length L, hich ill be the value for the parameter L in all the model formulas in Eqs. ( through (. We see that shear force roceedings of the Midest Section onference of the merican Societ for Engineering Education

14 V at the left end is equal to, the moment M and deflection at are zero. We ma let the slope and deflection at be and, respectivel. ppling the model formulas in Eqs. ( and ( to segment in Fig. (a, in that order, e rite ( L (h EI ( L L EI ( (i L ( L ( L L For equilibrium of the segment in Fig. (a, e rite F : V (j M : L LM (k We note that the slope D and deflection D at end D of segment D are zero. ppling the model formulas in Eqs. ( and ( to segment D in Fig. (b, in that order, e rite V L M L (l EI EI L ML L V (m EI The above six simultaneous Eqs. (h through (m ield ( I 5 I 9 ( I 8I ( 5 II V 9 ( I 8I M ( I I 9 I 8 I ( I II I EI I ( 9I 8I ( I I EI ( 9I 8I ( I I ( 9I 8I Using these values and appling the model formulas in Eqs. ( and (, e rite ( I 8 II I xl L EI EI I ( 9I 8I ( I 7I I 7I L L x L EI I ( 9I 8I ( I 5 I 9 ( I 8I ( I I I I EI I ( 9I 8I ( I 8 I I I EI I ( 9I 8I ( I I EI ( 9I 8I ( I 7I I 7I EI I ( 9 I 8 I ( I I ( 9I 8I roceedings of the Midest Section onference of the merican Societ for Engineering Education

15 5 hecking Obtained Results: The effort to obtain the solution for the problem in this example is algebraicall challenging. Naturall, it is desirable to check the preceding obtained results against a knon solution for the special case of I I I For such a special case, the preceding obtained results degenerate into the folloing: 7 5EI EI 8EI 7EI We find that these special results are indeed consistent ith those given at the end of textbooks. 9 III. n Effective pproach to Teaching 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 requires much less effort 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 five examples, arranged in order of increasing challenge, in Section II provide a variet of head-to-head comparisons beteen solutions b the traditional method of integration and those b the method of model formulas; and all of the solutions are, respectivel, in agreement. Thus, all solutions b the method of model formulas are naturall correct. 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 determining staticall indeterminate reactions and deflections of elastic beams in mechanics of materials: Teach the traditional 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 of the four model formulas in terms of singularit functions. Give students the heads-up on the folloing advantages 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, and the beam has linearl distributed loads not ending at its right end. Demonstrate solutions of several beam problems b the method of model formulas. ssess the solutions obtained (e.g., comparing ith solutions b another method. roceedings of the Midest Section onference of the merican Societ for Engineering Education

16 6 lthough solutions obtained b the method of model formulas are often more direct than those obtained b the method of integration, a one-page excerpt from the method of model formulas, such as that shon in Fig., must be made available to those ho used this method. Still, one ma remember that a list of formulas for slope and deflection of selected beams having a variet of supports and loading is also needed b persons ho use the method of superposition. In this regard, the method of model formulas is on a par ith the method of superposition. IV. oncluding 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 the beam that can readil be ritten. Selected applied loads are illustrated in Fig. (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 best taught to students as an alternative method, after the have learned one or more of the traditional methods. - This ne method enriches students stud and set of skills in determining reactions and deflections of beams, and it provides engineers ith a means to independentl check their solutions obtained using traditional methods. References. I.. Jong, n lternative pproach to Finding eam Reactions and Deflections: Method of Model Formulas, International Journal of Engineering Education, Vol. 5, No., pp. 65-7, 9.. S. Timoshenko and G. H. Macullough, Elements of Strength of Materials (rd Edition, Van Nostrand ompan, Inc., Ne York, NY, 99.. S. H. randall,. D. Norman, and T. J. Lardner, n Introduction to the Mechanics of Solids (nd Edition, McGra-Hill, Ne York, NY, 97.. R. J. Roark and W.. Young, Formulas for Stress and Strain (5th Edition, McGra-Hill, Ne York, NY, F. L. Singer and. tel, Strength of Materials (th Edition, Harper & Ro, Ne York, NY, tel and J. Kiusalaas, Mechanics of Materials, rooks/ole, acific Grove,,. 7. J. M. Gere, Mechanics of Materials (6th Edition, rooks/ole, acific Grove,,. 8. F.. 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, 8.. H. M. Westergaard, Deflections of eams b the onjugate eam Method, Journal of the Western Societ of Engineers, Vol. XXVI, No., pp , 9.. H. T. Grandin, and J. J. Rencis, Ne pproach to Solve eam Deflection roblems Using the Method of Segments, roceedings of the 6 SEE nnual conference & Exposition, hicago, IL, 6.. I.. Jong, Deflection of a eam in Neutral Equilibrium à la onjugate eam Method: Use of Support, Not oundar, onditions, 7 th SEE Global olloquium on Engineering Education, ape Ton, South frica, October 9-, 8. roceedings of the Midest Section onference of the merican Societ for Engineering Education

17 7 ING-HNG JONG Ing-hang Jong is rofessor of Mechanical Engineering at the Universit of rkansas. He received his SE in 96 from the National Taian Universit, his MSE in 96 from South Dakota School of Mines and Technolog, and his h.d. in Theoretical and pplied Mechanics in 965 from Northestern Universit. Dr. Jong as hair of the Mechanics Division, SEE, in He received the rchie Higdon Distinguished Educator ard in 9. WILLIM T. SRINGER William T. Springer is ssociate rofessor of Mechanical Engineering at the Universit of rkansas. He received his SME in 97 from the Universit of Texas at rlington, his MSME in 979 from the Universit of Texas at rlington, and his h.d. in Mechanical Engineering in 98 from the Universit of Texas at rlington. Dr. Springer is active in SME here he received the Dedicated Service ard in 6 and attained Fello Grade in 8. RIK J. OUVILLION Rick J ouvillion is ssociate rofessor of Mechanical Engineering at the Universit of rkansas. He received his SME in 975 from the Universit of rkansas. fter ½ ears in industr, he returned to school, received his MSME from Georgia Tech in 978, and his h.d. from Georgia Tech in 98. He is active in SHRE and as elected an SME fello in. He as chosen for the Universit of rkansas Teaching cadem in 5. roceedings of the Midest Section onference of the merican Societ for Engineering Education