Determining Deflections of Hinge-Connected Beams Using Singularity Functions: Right and Wrong Ways

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1 Determining Deflections of Hinge-Connected Beams Using Singularit Functions: Right and Wrong Was Ing-Chang Jong Universit of rkansas bstract When the method of double integration is used to determine deflections, as well as staticall indeterminate reactions at supports, of a beam in echanics of aterials, one has the option of using singularit functions to account for all loads on the entire beam in formulating the solution. This option is an effective wa and a right wa to solve the problem if the beam is a single piece of elastic bod. However, this option becomes a wrong wa to do it if one fails to heed the existence of discontinuit in the slope of the beam under loading. Beginners tend to have a misconception that singularit functions are a powerful mathematical tool, which can allow one to blaze the loads on the entire beam without the need to divide it into segments. It is pointed out in this paper that hinge-connected beams are a pitfall for unsuspecting beginners. The paper reviews the sign conventions for beams and definitions of singularit functions, and it includes illustrations of both right and wrong was in solving a problem involving a hinge-connected beam. It is aimed at contributing to the better teaching and learning of mechanics of materials. I. Introduction There are several established methods for determining deflections of beams in mechanics of materials. The include the following: -9 (a) method of double integration (with or without the use of singularit functions), (b) method of superposition, (c) method using moment-area theorems, (d) method using Castigliano s theorem, (e) conjugate beam method, and (f) method using general formulas. Naturall, there are advantages and disadvantages in using an of the above methods. B and large, the method of double integration is the commonl used method in determining slopes and deflections, as well as staticall indeterminate reactions at supports, of beams. Without using singularit functions, the method of double integration has a disadvantage, because it requires division of a beam into segments for individual studies, where the division is dictated b the presence of concentrated forces or moments, or b different flexural rigidities in different segments. Readers, who are familiar with mechanics of materials, ma skip the refresher on the rudiments included in the earl part of this paper. Sign Convention. 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 of beams. segment of beam ab having a constant flexural rigidit EI is shown in Fig.. Note that we adopt the positive directions of the shear forces, moments, and distributed loads as indicated. "Proceedings of the 006 idwest Section Conference of the merican Societ for

2 Fig. Positive directions of shear forces, moments, and loads s in most textbooks for mechanics of materials, notice in Fig. the following conventions: -6 (a) positive shear force is one that tends to rotate the beam segment clockwise (e.g., V a at the left end a, and V b at the right end b). (b) positive moment is one that tends to cause compression in the top fiber of the beam (e.g., a at the left end a, b at the right end b, and the applied moment K tending to cause compression in the top fiber of the beam just to the right of the position where the moment K acts). (c) positive concentrated force applied to the beam is one that is directed downward (e.g., the applied force P). (d) a positive distributed load is one that is directed downward (e.g., the uniforml distributed load with intensit, and the linearl varing distributed load with highest intensit w ). w0 Fig. Positive deflections and positive slopes of beam ab The positive directions of deflections and slopes of the beam are defined as illustrated in Fig.. s in most textbooks for mechanics of materials, notice in Fig. the following conventions: -6 (i) positive deflection is an upward displacement (e.g., a at position a, and b at position b). (ii) positive slope is a counterclockwise rotation (e.g., θ a at position a, and θ b at position b). Singularit functions. Note that the argument of a singularit function is usuall enclosed b angle brackets (i.e., < >), while the argument of a regular function is enclosed b rounded parentheses [i.e., ( )]. The relations between these two functions are defined as follows: 7, 8 n n < x a > ( x a) if x a 0 and n 0 n < x a > 0 if x a< 0 or n< 0 () () "Proceedings of the 006 idwest Section Conference of the merican Societ for

3 x n n+ < x a > dx x a if n 0 n + < > () x x a n dx x a n+ < > < > n< if 0 Based on the sign conventions and the singularit functions defined above, we ma write the loading function q, the shear force V, and the bending moment for the beam ab in Fig. as (4) follows:, q V < x> + < x> P< x x > + K< x x > a a P K w w < x x > < x x > 0 0 w w L xw V V < x> + < x> P< x x > + K< x x > 0 0 a a P K w w < x x > < x x > 0 w w ( L xw) V < x> + < x> P< x x > + K< x x > 0 0 a a P K w < w x x > x x ( ) < > 0 w w 6 L xw (5) (6) (7) II. nalsis of a Hinge-Connected Beam: Right and Wrong Was ost textbooks for mechanics of materials or mechanical design do not sufficientl warn their readers that singularit functions can be elegantl used to overcome discontinuities in the various loads acting on the entire beam [such as those shown in Eqs. (5), (6), and (7) for the loads shown in Fig. ], but the cannot blaze the various loads for the entire beam when the beam has one or more discontinuities in its slope when the loads are applied to act on it. In fact, singularit functions cannot be above the rules of mathematics that require a function to have continuous slopes in a domain if it is to be integrated or differentiated in that domain. Here, the beam is the domain. If a beam is composed of two or more segments that are connected b hinges (e.g., a Gerber beam), then the beam has discontinuous slopes at the hinge connections when loads are applied to act on it. In such a case, the deflections and an staticall indeterminate reactions must be analzed b dividing the beam into segments, each of which must have no discontinuit in slope. Otherwise, erroneous results will be reached. Example. combined beam (Gerber beam) having a constant flexural rigidit EI is loaded and supported as shown in Fig.. Show a wrong wa to use singularit functions to attempt a solution for the vertical reaction force and the reaction moment at. Fig. Fixed-ended beam with a hinge connector "Proceedings of the 006 idwest Section Conference of the merican Societ for

4 4 Wrong wa. For illustrative purpose, let us first show how a wrong wa ma be used b an unsuspecting person in tring to solve the problem and reaching wrong results as follows: Fig. 4 Free-bod diagram with assumption of positive reaction forces and moments Since this person would use singularit functions to blaze the loading for the entire beam, the loading function q, the shear force V, and the bending moment for the entire beam would be as follows: q < x > + < x > P< x L> Double integration of the last equation ields V < x > + < x > P< x L> 0 0 EI < x> + < x> P< x L> 0 EI < x> + < x> P< x L> + C EI < x> + < x> P< x L>+ C x+ C 6 6 Imposition of boundar conditions ields (0) 0: 0 C (a) (0) 0: 0 C (b) ( L) 0: ( L ) 0: 0 ( ) (9 ) L + L +C (c) L + L + C L + C (d) (9 ) (7 ) ( ) Solution of simultaneous Eqs. (a) through (d) ields C 0 C 0 7 P 9 7 Consistent with the defined sign conventions, this unsuspecting person would report 7 P 9 7 Note that these two answers are wrong because we can refer to Fig. 4 and show that the do not satisf the fact that the magnitude of moment B 0 at the hinge at B; i.e., 7 B + L + P ( L ) "Proceedings of the 006 idwest Section Conference of the merican Societ for

5 5 Example. combined beam (Gerber beam) having a constant flexural rigidit EI is loaded and supported as shown in Fig.. Show the right wa to use singularit functions to determine for this beam (a) the vertical reaction force and the reaction moment at, (b) the deflection B of the hinge at B, (c) the slopes θ BL and θ BR just to the left and just to the right of the hinge at B, respectivel, and (d) the slope θ C and the deflection at C. C Fig. Fixed-ended beam with a hinge connector (repeated) Right wa. This beam is staticall indeterminate to the first degree. Nevertheless, because of the discontinuit in slope at the hinge connection B, this beam needs to be divided into two segments B and for analsis in the solution, where no discontinuit in slope exists within each segment. Fig. 5 Free-bod diagram for segment B and its deflections The loading function qb, the shear force VB, and the bending moment as shown in Fig. 5, are q < x > + < x > Double integration of the last equation ields B B V < x > + < x > B B 0 EI < x> + < x> 0 EI < x> + < x> + C B EIB < x> + < x> + C x+ C 6 B for the segment B, Fig. 6 Free-bod diagram for segment and its deflections "Proceedings of the 006 idwest Section Conference of the merican Societ for

6 6 The loading function q, the shear force V, and the bending moment, as shown in Fig. 6, are q B < x > P< x L> for the segment Double integration of the last equation ields V B < x > P< x L> 0 0 EI B < x> P< x L> EI B< x> P< x L> + C EI B < x> P< x L> + C x+ C 6 6 Imposition of boundar conditions ields 4 (0) 0 : B 0 C (a) (0) 0 : 0 C (b) B ( L) (0) : B ( ) 0 : L ( ) 0 : L L + L C4 (c) 6 0 B (4 ) L + C (d) B L + C L + C (e) (8 ) ( ) 4 Imposition of equations of static equilibrium for segment B ields + Σ B 0 : 0 L (f) + Σ 0: B 0 (g) F Solution of simultaneous Eqs. (a) through (g) ields C 0 C 0 5P 8 B 5P 8 C 8 Consistent with the defined sign conventions, we report that C P Substituting the above solutions into foregoing equations for EI, EI B, and EI, respectivel, we write "Proceedings of the 006 idwest Section Conference of the merican Societ for

7 7 EI (0) 5 B EI C θ EI BL EIB ( L) L L C B 5 54EI 5 θ BL 6EI EIθ EI (0) C BR 8 θ BR 8EI 6EI C ( ) + θ C EIθ EI L B L C EI C ( ) C EI EI L B L C L C EI Based on the preceding solutions, the deflections of the combined beam D ma be illustrated as shown in Fig. 7. Fig. 7 Deflections of the beam D Concluding Remarks This paper provides a refresher on the sign conventions for beams and definitions of singularit functions. Beginners in mechanics of materials are usuall not sufficientl warned about the limitations of what singularit functions can do. Students tend to have a misconception that singularit functions are a powerful mathematical tool, which can allow them to blaze the loads on the entire beam without the need to divide it into segments for analsis. It is pointed out in this paper that hinge-connected beams are a pitfall for unsuspecting beginners. The paper includes two illustrative examples to demonstrate both wrong and right was in using singularit functions to solve a problem involving a hinge-connected beam. It is emphasized that singularit functions cannot be above the rules of mathematics that require a function to have continuous slopes in a domain if it is to be integrated or differentiated in that domain. In mechanics of materials, the beam is the domain. If a beam is composed of two or more segments that are connected b hinges (e.g., a Gerber beam), then the beam has discontinuous slopes at the hinge connections when loads are applied to act on it. In general, the deflections and an staticall indeterminate reactions must be analzed b dividing the beam into segments, as needed, each of "Proceedings of the 006 idwest Section Conference of the merican Societ for

8 8 which must have no discontinuit in slope. Otherwise, erroneous results will be reached. This paper is aimed at contributing to the better teaching and learning of mechanics of materials. References. Westergaard, H.., Deflections of Beams b the Conjugate Beam ethod, Journal of the Western Societ of Engineers, Volume XXVI, Number, 9, pp Timoshenko, S., and G. H. accullough, Elements of Strength of aterials, Third Edition, D. Van Nostrand Compan, Inc., New York, NY, 949, pp Singer, F. L., and. Ptel, Strength of aterials, Fourth Edition, Harper & Row, Publishers, Inc., New York, NY, 987, pp Beer, F. P., E. R. Johnston, Jr., and J. T. DeWolf, echanics of aterials, Fourth Edition, The cgraw-hill Companies, Inc., New York, NY, Ptel,., and J. Kiusalaas, echanics of aterials, Brooks/Cole, Pacific Grove, C, Gere, J.., echanics of aterials, Sixth Edition, Brooks/Cole, Pacific Grove, C, Shigle, J. E., echanical Engineering Design, Fourth Edition, cgraw-hill Compan, New York, NY, 98, pp Crandall, S. H., C. D. Norman, and T. J. Lardner, n Introduction to the echanics of Solids, Second Edition, cgraw-hill Compan, New York, NY, 97, pp Jong, I. C., J. J. Rencis, and H. T. Grandin, New approach to nalzing Reactions and Deflections of Beams: Formulation and Examples, IECE006-90, Proceedings of 006 SE International echanical Engineering Congress and Exposition, November 5-0, 006, Chicago, IL, U.S.. ING-CHNG JONG Ing-Chang Jong serves as Professor of echanical Engineering at the Universit of rkansas. He received a BSCE in 96 from the National Taiwan Universit, an SCE in 96 from South Dakota School of ines and Technolog, and a Ph.D. in Theoretical and pplied echanics in 965 from Northwestern Universit. He was Chair of the echanics Division, SEE, in His research interests are in mechanics and engineering education. "Proceedings of the 006 idwest Section Conference of the merican Societ for