AC : ANALYSIS OF STATICALLY INDETERMINATE REACTIONS AND DEFLECTIONS OF BEAMS USING MODEL FORMULAS: A NEW APPROACH

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1 C 7-11: NYSIS OF STTICY INDETERMINTE RECTIONS ND DEFECTIONS OF EMS USING MODE FORMUS: NEW PPROCH Ing-Chang Jong, Universit of rkansas Ing-Chang Jong serves as 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 as Chair of the Mechanics Division, SEE, in His research interests are in mechanics and engineering education. Joseph Rencis, Universit of rkansas Joseph J. Rencis is Professor and Head of the Department of Mechanical Engineering at the Universit of rkansas in Faetteville. From 1985 to 4 he as in the Mechanical Engineering Department at the Worcester Poltechnic Institute. His research focuses on the development of boundar and finite element methods for analzing solid, heat transfer and fluid mechanics problems ith a focus on multi-scale modeling. He serves on the editorial board of Engineering nalsis ith oundar Elements and is associate editor of the International Series on dvances in oundar Elements. He is currentl the Secretar/Treasurer of the SEE Mechanical Engineering Division and Vice Chair of the SME Mechanical Engineering Department Heads Committee. He as Chair of the SEE Mechanics Division, 1999-, received aard as the SEE Ne England Section Teacher of the Year, and is an SME fello. In 4 he received the SEE Ne England Section Outstanding eader ard and in 6 the SEE Mechanics Division James. Meriam Service ard. He received his.s. from the Milaukee School of Engineering in 198, an M.S. from Northestern Universit in 198, and a Ph.D. from Case Western Reserve Universit in V-mail: ; jjrencis@uark.edu. merican Societ for Engineering Education, 7 Page 1.4.1

2 nalsis of Staticall Indeterminate Reactions and Deflections of eams Using Model Formulas: Ne pproach bstract This paper is intended to share ith educators and practitioners in mechanics a ne approach that emplos a set of four model formulas in analzing staticall indeterminate reactions at supports, as ell as the slopes and deflections, of beams. The model formulas, in algebraic form, are derived using singularit functions. The are expressed in terms of (a) flexural rigidit of the beam; (b) slopes and deflections, as ell as shear forces and bending moments, at both ends of the beam; and (c) applied loads on the beam. The tpes of applied loads include: (i) concentrated force and moment; (ii) uniforml distributed moment; and (iii) linearl varing distributed force. Thus, these model formulas are applicable to most problems encountered in the teaching and learning of mechanics of materials, as ell as in practice. s a salient feature, this ne approach allos one to treat reactions at supports, even not at the ends of a beam, simpl as concentrated forces or moments, here corresponding boundar conditions at the points of supports are imposed using also the model formulas. This feature allos one to readil determine staticall indeterminate reactions at supports, as ell as slopes and deflections at an positions, of beams. beam needs to be divided into segments for analsis onl hen it has discontinuit in slope or in flexural rigidit. Several examples are provided to illustrate this ne approach. I. Introduction There are different ell-knon methods for determining deflections of beams in mechanics of 1 1 materials. These methods ma include the folloing: (a) method of double integration (ith or ithout the use of singularit functions), (b) method of superposition, (c) method using moment-area theorems, (d) method using Castigliano s theorem, and (e) conjugate beam method. This paper extends an earlier stud on method of segments 11 b using singularit functions and model formulas. s a result, the proposed ne approach allos a considerable reduction in the number of segments required in the stud. This ne approach makes available an effective method for mechanics educators and practitioners hen it comes to determining reactions and deflections of beams. It is aimed at contributing to the enrichment of one s learning experience and to provide a means for independent checking on solutions obtained b other methods. The paper goes over the description of sign conventions and derives four model formulas for the slope and deflection of a beam segment having a constant flexural rigidit and carring a variet of commonl applied loads. These formulas, derived using singularit functions, form the basis for a ne approach to solving problems involving reactions and deflections of beams. In contrast to the method of segments, 11 the proposed ne approach does not have to divide a beam into multiple segments even if the beam has multiple concentrated loads or simple supports not at its Page 1.4.

3 ends. pplication of these model formulas is direct and requires no further integration or riting of continuit equations. The model formulas can readil be extended to the analsis of beams having discontinuit in slope (e.g., at hinge connections) or in flexural rigidit (e.g., in stepped segments). It can solve both staticall determinate and staticall indeterminate beam problems. II. Sign Conventions for eams In the analsis of beams, it is important to adhere to the generall agreed positive and negative signs for loads, shear forces, bending moments, slopes, and deflections. The free-bod diagram for a beam ab having a constant flexural rigidit EI and carring loads is shon in Fig. 1. The positive directions of shear forces Va and Vb, moments Ma and Mb, at ends a and b of the beam, the concentrated force P and concentrated moment K, as ell as the distributed loads, are illustrated in this figure. Figure 1. Positive directions of shear forces, moments, and applied loads In general, e have the folloing sign conventions for shear forces, moments, and applied loads: shear force is positive if it acts upard on the left (or donard on the right) face of the beam element (e.g., Va at the left end a, and Vb at the right end b in Fig. 1). 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 Mb at the right end b in Fig. 1). 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 and the uniforml distributed moment ith intensit m in Fig. 1). concentrated force or a distributed force applied to the beam is positive if it is directed donard (e.g., the concentrated force P, the linearl varing distributed force ith intensit on the left side and 1 on the right side in Fig. 1, here the distribution becomes uniform if ). 1 Furthermore, e adopt the folloing sign conventions for deflection and slope of a beam: positive deflection is an upard displacement. positive slope is a counterclockise angular displacement. Page 1.4.

4 III. Derivation of Model Formulas n beam element of differential idth dx at an position x ma be perceived to have a left face and a right face. Using singularit functions, 8-1 e ma rite, for the beam ab in Fig. 1, the loading function q, shear force V, and bending moment M acting on the left face of the beam element at an position x for this beam as follos: qv x M x Pxx Kxx xx 1 1 a a P K 1 x x 1 1 m x x m x 1 1 VVax Max PxxP KxxK xx 1 xx mxxm ( x) M V x M x Pxx Kxx xx x x m x x 1 1 a a P K 1 1 m 6( x) 1 (1) () () It is important to note that Eqs. (1) through () are ritten for the beam in the range x <, and e have x even though x at the right end of beam. the definition of singularit functions, the value of x is zero henever x, regardless of the value of n. n 1 Therefore, values of the terms Vb x Mb x, as ell as the integrals of these terms, are alas equal to zero for the beam. This is the reason h these to terms are trivial and ma simpl be omitted in the expression for the loading function q in Eq. (1). Noting that EI is the flexural rigidit, is the deflection, is the slope, is the second derivative of ith respect to x for an section of the beam ab, and EI M, e rite 4 EI V x M x Pxx Kxx xx x x m x x 1 1 a a P K 1 1 m 6( x) 1 1 EI V x M x P x x + K x x x x 1 1 a a P K xx m< xxm> C1 4( x) EI 1 V x 1 M x 1 P x x 1 K x x 6 6 xx 1 m xx CxC 1( ) 6 a a P K 1 5 m> 1 x x x 4 4 (4) (5) (6), re- The slope and deflection of the beam in Fig. 1 at its left end a (i.e., at x = ) are a and spectivel. Imposition of these to boundar conditions on Eqs. (5) and (6) ields a C C EI (7) 1 a EI (8) a Page 1.4.4

5 Substituting Eqs. (7) and (8) into Eqs. (5) and (6), e obtain the model formulas for the slope and deflection, at an position x, of the beam ab in Fig. 1 as follos: V M P K x x xx xx x x EI EI EI EI m xx xx 4EI x EI a a 1 a P K 1 4 m ( ) V M P K x x x xx xx EI EI m xx xx 1EI 6 a a 4 a a P K 1 5 m> ( x) EI 4EI x x (9) (1) letting x = in Eqs. (9) and (1), e obtain the model formulas for the slope b and deflection at the right end b of the beam ab as follos: b Va Ma P a ( xp) K ( xk ) b EI EI EI EI 1 m ( x) ( xm) 4EI EI V M P K b a x x EI EI m ( x) ( xm) 1EI a a a ( P ) ( K ) (11) (1) IV. pplications of Model Formulas The set of four model formulas given b Eqs. (9) through (1) ma be used as the basis upon hich to formulate a ne approach to analzing staticall indeterminate reactions at supports, as ell as the slopes and deflections, of beams. The beams ma carr a variet of applied loads, as illustrated in Fig. 1. Note that in the model formulas is a parameter representing the total length of the beam segment, to hich the model formulas are to be applied. These formulas have alread taken into account the boundar conditions of the beam at its ends. Furthermore, this approach allos one to treat reactions at interior supports (those not at the ends of the beam) as applied concentrated forces or moments. ll one has to do is to simpl impose the additional corresponding boundar conditions at the interior supports for the beam segment. Thus, the ne approach allos one to readil determine staticall indeterminate reactions as ell as slopes and deflections of beams. beam needs to be divided into separate segments for analsis onl if (a) it is a combined beam (e.g., Gerber beam) having discontinuit in slope at hinge connections beteen segments, and (b) it contains segments of different flexural rigidities. The ne approach proposed in this paper can best be understood ith illustrations. Therefore, simple as ell as more challenging problems are included in the folloing examples. Page 1.4.5

6 Example 1. cantilever beam ith a constant flexural rigidit EI and a length is acted on b three concentrated forces of magnitudes P, P, and P as shon in Fig.. For this beam, determine the slope and deflection at its free end. Figure. Cantilever beam carring three concentrated forces Solution. There is no need to divide the beam into segments for stud. t end, the moment M is zero and the shear force V is P. t end, the slope and deflection are both zero. Since e have multiple concentrated forces acting on the beam, e need to appl the concentrated force term (the term containing P) multiple times in all model formulas. ppling the model formulas in Eqs. (11) and (1), successivel, to this beam, e rite P P P EI EI EI P P P The above to simultaneous equations, containing the to unknons and 1P 67P 9EI 81EI Consistent ith the defined sign conventions, e report that, ield 1P 9EI 67P 81EI Example. The right end of a fix-ended beam, hich has a constant flexural rigidit EI and a length, is shifted upard b an amount as shon in Fig.. For such a relative vertical shifting of supports, determine (a) the vertical reaction force and the reaction moment M developed at the left end, (b) the deflection of the beam at an position x. Figure. Relative vertical shifting of supports in a fix-ended beam Solution. This beam is staticall indeterminate to the second degree. t the fixed support, the deflection and slope are zero. t the fixed support, the deflection is, but the slope is zero. ppling the model formulas in Eqs. (11) and (1), successivel, to this beam, e rite Page 1.4.6

7 M EI EI M EI The above to simultaneous equations, containing the to unknons 1 6 EI M E I Consistent ith the defined sign conventions, e report that 1 EI Substituting the obtained values of and M M x x EI and M, ield M into the model formula in Eq. (1), e rite x x x x Example. cantilever beam ith a constant flexural rigidit EI and carring a uniforml distributed moment of intensit m over its entire length is shon in Fig. 4. Determine (a) the slope and deflection at the free end, (b) the deflection of the beam at an position x. Figure 4. Cantilever beam carring a uniforml distributed moment Solution. The free-bod diagram of the cantilever beam, hich is in equilibrium as shon in Fig. 5, indicates that the beam has onl a counterclockise reaction moment of magnitude m at its end besides a uniforml distributed moment of intensit over its entire length. m Figure 5. Free-bod diagram of the cantilever beam We note that the deflection, slope, and the shear force at end in Fig. 5 are all zero. t end, the moment M and shear force are both zero. ppling the model formulas in Eqs. (11) and (1), successivel, to this beam and noting that x m =, e rite ( m ) m EI EI ( ) ( m ) m ( ) EI Page 1.4.7

8 The above to simultaneous equations, containing the to unknons and m m EI EI Consistent ith the defined sign conventions, e report that, ield m EI Substituting the obtained values of and m EI into the model formula in Eq. (1), e rite m m m x x x x EI ( ) m x x 6 EI ( ) Example 4. cantilever beam ith a constant flexural rigidit EI and a length is propped b a linear tension-compression spring of modulus k, and it carries a concentrated moment K at its midpoint C as shon in Fig. 6. Determine the slope and deflection at its left end. Figure 6. Cantilever beam propped b a linear spring and carring a concentrated moment Solution. t end of this beam, the moment M is zero and shear force V = k, hich is based on the initial assumption that is upard and the linear spring force of k acts donard at end. t end, the slope and deflection are both zero. Note that e need to set the parameter in the model formulas in Eqs. (11) and (1) equal to for this beam. etting x = and appling Eqs. (11) and (1), successivel, to this beam, e rite K ( k )( ) K ( ) EI EI ( k )( ) K ( ) ( ) EI The above to simultaneous equations, containing the to unknons and K( EI k ) 9K EI EI k EI k ( 8 ) ( 8 ) Consistent ith the defined sign conventions, e report that, ield K( EI k ) EI(EI 8 k ) 9K EI k ( 8 ) Page 1.4.8

9 Example 5. beam, hich has a constant flexural rigidit EI, a roller support at, a roller support at C, a fixed support at, and a length of, carries a linearl distributed load ith maximum intensit 1 at as shon in Fig. 7. Determine (a) the vertical reaction force and the slope at, (b) the vertical reaction force C and the slope at C. C Figure 7. eam supported b to rollers and a fixed support Solution. This beam is staticall indeterminate to the second degree. There is no need to divide the beam into to segments for analsis in the solution b the proposed ne approach. We can simpl treat the vertical reaction force C at C as an unknon applied concentrated force, directed upard, and regard the beam as one that has a total length of, hich is to be used as the value for the parameter in the model formulas in Eqs. (9) through (1). The boundar conditions of this beam reveal that the moment M and the deflection at are both zero, the slope and deflection at are also both zero, and the deflection C at C is zero. The shear force at the left end is the vertical reaction force at, hich ma be assumed to be acting upard. ppling Eqs. (11) and (1) to the entire beam and Eq. (1) for imposing that C at C, in that order, e rite ( ) C 1 ( ) ( ) EI EI 4EI ( ) C 1 4 ( ) ( ) ( ) 6 EI 1 EI The above three simultaneous equations, containing the three unknons,, and C 7 4EI 14 Consistent ith the defined sign conventions, e report that C, ield 1 7 4EI C The slope C is simpl evaluated at C, hich is located at x =. ppling the model formula in Eq. (9) and utilizing the preceding solutions for and, e rite C EI 4EI 7 EI 1EI 1EI 1 C Page 1.4.9

10 Example 6. combined beam (Gerber beam), ith a constant flexural rigidit EI, fixed supports at its ends and D, a hinge connection at, and carring a concentrated moment K at C, is shon in Fig. 8. Determine (a) the vertical reaction force and the reaction moment M at, (b) the deflection of the hinge at, (c) the slopes and R just to the left and just to the right of the hinge at, respectivel, and (d) the slope C and the deflection C at C. Figure 8. Fix-ended beam ith a hinge connector Solution. This beam is staticall indeterminate to the first degree. ecause of the discontinuit in slope at the hinge connection, this beam needs to be divided into to segments and D for analsis in the solution, as shon in Figs. 9 and 1, here the deflected shapes are shon to highlight the discontinuit in slope at ; i.e., R. The boundar conditions of this beam reveal that slope and deflection at and D are all equal to zero. Figure 9. Free-bod diagram for segment and its deflection ppling the model formulas in Eqs. (11) and (1), successivel, to segment, as shon in Fig. 9, e rite M EI EI (a) M EI (b) For equilibrium of segment in Fig. 9, e rite : (c) F M : M (d) Figure 1. Free-bod diagram for segment D and its deflection Page 1.4.1

11 ppling the model formulas in Eqs. (11) and (1), successivel, to segment D, as shon in Fig. 1, e rite ( ) K R () (e) EI EI ( ) (f) EI K R ( ) ( ) For equilibrium of segment D in Fig. 1, e rite : D (g) F M : K D MD (h) The above eight simultaneous equations, in eight unknons, ield K K K K M 4 EI K K R D MD Consistent ith the defined sign conventions, e report that K M K K 6 EI K R 4EI The position C is located at x =, as shon in Fig. 1. ppling the model formulas in Eqs. (9) and (1), successivel, to the segment D in this figure and utilizing the preceding solutions for and, e rite R EI C R ( ) K 4EI K C R ( ) 1EI K C 4EI C K 1EI ased on the preceding solutions, the deflections of the combined beam D ma be illustrated as shon in Fig. 11. Figure 11. Deflection of the combined beam Page

12 Example 7. stepped beam C carries a uniforml distributed load as shon in Fig. 1, here the segments and C have flexural rigidities EI1 and EI, respectivel. Determine (a) the deflection at, (b) the slopes,, and C at,, and C, respectivel. Figure 1. Stepped beam carring a uniforml distributed load Solution. ecause of the discontinuit in flexural rigidities at, this beam needs to be divided into to segments and C for analsis in the solution. The boundar conditions of this beam reveal that the deflections at and C are zero. Figure 1. Free-bod diagram for segment ppling the model formulas in Eqs. (11) and (1), successivel, to segment and noting that 1, as shon in Fig. 1, e rite 1 (a) EI 4EI EI1 1 ( ) (b) 1 For equilibrium of segment in Fig. 1, e rite F : 1 (c) M : 1 1 M (d) Figure 14. Free-bod diagram for segment C ppling the model formulas in Eqs. (11) and (1), successivel, to segment C, as shon in Fig. 14, e rite Page 1.4.1

13 M C (e) EI EI M (f) EI For equilibrium of segment C in Fig. 14, e rite : C (g) F M : M (h) The above eight simultaneous equations, in eight unknons, ield ( ) C M ( ) ( ) ( ) ( ) [4 I ( 5 ) I] ( 1 ) EI1I [4 I ( 5 8 ) I ] ( 1 ) EI1I [4 I ( 5 ) I ] ( 1 ) EI1I C [ ( ) I ( 5 ) I ] ( 1 ) EI1I The above solutions have been assessed and numericall verified to be in agreement ith the ansers that ere independentl obtained for a problem involving the same beam but being solved using a different method the method of segments. 11 V. dvantages of Model Formulas pproach over Method of Segments The proposed approach using model formulas represents a significant extension of the method of segments. 11 In comparison ith the method of segments, this approach embraces several major advantages b alloing the folloing: Multiple concentrated and distributed loads. single beam segment is alloed to carr simultaneousl an number of concentrated forces, concentrated moments, distributed forces, and distributed moments. Multiple supports anhere of the beam. single beam segment is alloed to have multiple supports that are (a) rigid (e.g., roller or hinge), not at the ends; (b) non-rigid (e.g., tension- Page 1.4.1

14 compression spring or torsional spring), not at the ends; and (c) either rigid (e.g., roller, hinge, or fixed) or non-rigid, at the ends. Prescribed linear or angular displacements of the beam at an of its supports. single beam segment is alloed to have a prescribed translational or rotational displacement at an of its supports, regardless of the support being at the end or not at the end of the beam. Significant reduction in required number of segments into hich a beam must be divided. Most beam problems can be solved b the model formulas approach using onl one segment. Division of a beam into to or more segments is required onl hen the beam has discontinuit in slope (e.g., at hinge connections) or in flexural rigidit (e.g., in stepped segments). Significant reduction in number of equations and unknons generated in the solution. The number of equations and unknons in a beam problem increases as the required number of segments is increased. Since the model formulas approach allos most beam problems to be modeled ith a single segment (except in the case of a beam having discontinuit in slope or in flexural rigidit), the number of equations and unknons is significantl reduced. VI. Conclusion This paper is presented to share ith educators and practitioners in mechanics a ne approach that emplos a set of four model formulas in solving problems involving staticall indeterminate reactions at supports, as ell as the slopes and deflections, of beams. These formulas, derived using singularit functions, provide the material equations, besides the equations of static equilibrium, for the solution of the problem. The are expressed in terms of (a) flexural rigidit of the beam; (b) slopes and deflections, as ell as shear forces and bending moments, at both ends of the beam; and (c) applied loads on the beam. Tpical applied loads are illustrated in Fig. 1, hich include (i) concentrated force and concentrated moment; (ii) uniforml, as ell as linearl varing, distributed force, and (iii) uniforml distributed moment. The case of multiple concentrated forces acting on a beam is illustrated in Example 1, here the term containing P in the model formulas has been applied multiple times to account for the multiple concentrated forces. For other tpes of multiple loads on the beam, one ma similarl appl multiple times the appropriate terms in the model formulas. s a salient feature, the proposed ne approach allos one to treat unknon reactions at supports not at the ends of a beam simpl as concentrated forces or moments. The boundar conditions at such supports are readil imposed using also the model formulas. beam needs to be divided into separate segments for analsis onl hen it has discontinuit in slope or in flexural rigidit. Finall, one should remember that the parameter in the model formulas represents the total length of the beam segment, to hich the formulas are to be applied. VII. References 1. Westergaard, H. M., Deflections of eams b the Conjugate eam Method, Journal of the Western Societ of Engineers, Volume XXVI, Number 11, 191, pp Page

15 . Timoshenko, S., and G. H. MacCullough, Elements of Strength of Materials, Third Edition, D. Van Nostrand Compan, Inc., Ne York, NY, Singer, F.., and. Ptel, Strength of Materials, Fourth Edition, Harper & Ro, Publishers, Inc., Ne York, NY, eer, F. P., E. R. Johnston, Jr., and J. T. DeWolf, Mechanics of Materials, Fourth Edition, The McGra-Hill Companies, Inc., Ne York, NY, Jong, I. C., Effective Teaching and earning of the Conjugate eam Method: Snthesized Guiding Rules, Session 468, Mechanics Division, Proceedings of the 4 SEE nnual Conference & Exposition, Salt ake Cit, UT, June -, Ptel,., and J. Kiusalaas, Mechanics of Materials, rooks/cole, Pacific Grove, C,. 7. Gere, J. M., Mechanics of Materials, Sixth Edition, rooks/cole, Pacific Grove, C, Shigle, J. E., Mechanical Engineering Design, Fourth Edition, McGra-Hill Compan, Ne York, NY, 198, pp Norton, R.., Machine Design: n Integrated pproach, Third Edition, Pearson Prentice Hall, Upper Saddle River, NJ, 6, pp Crandall, S. H., C. D. Norman, and T. J. ardner, n Introduction to the Mechanics of Solids, Second Edition, McGra-Hill Compan, Ne York, NY, 197, pp Grandin, H. T., and J. J. Rencis, Ne pproach to Solve eam Deflection Problems using the Method of Segments, Session 1568, Mechanics Division, Proceedings of the 6 SEE nnual Conference & Exposition, Chicago, I, June 18-1, 6. Page