TORSION EQUATIONS FOR ATERA BUCKING NS TRAHAIR RESEARCH REPORT R96 Jul 6 ISSN 8-78 SCHOO OF CIVI ENGINEERING
SCHOO OF CIVI ENGINEERING TORSION EQUATIONS FOR ATERA BUCKING RESEARCH REPORT R96 NS TRAHAIR Jul 6 ISSN 8-78
Copright Notice School of Civil Engineering, Research Report R96 Torsion Euations for ateral Buckling NS Trahair Jul 6 ISSN 8-78 This publication ma be redistributed freel in its entiret and in its original form without the consent of the copright owner. Use of material contained in this publication in an other published works must be appropriatel referenced, and, if necessar, permission sought from the author. Published b: School of Civil Engineering Sdne NSW 6 Australia This report and other Research Reports published b the School of Civil Engineering are available at http://sdne.edu.au/civil School of Civil Engineering Research Report r96 Page
ABSTRACT A torsion differential euation previousl used for analsing the elastic lateral buckling of simpl supported doubl smmetric beams with distributed loads acting awa from the centroidal ais omits an epected term and includes an unepected term. A different euation is derived b two different methods, either b using the calculus of variations with the second variation of the total potential, or b considering the euilibrium of the deflected and twisted beam. Four different methods are used to find solutions for the elastic buckling of beams with uniforml distributed loads. Two of these solve the differential euations numericall, either b using a computer program based on the method of finite integrals, or b making hand calculations with a single term approimation of the buckled shape. These methods produce different solutions for the two torsion differential euations. The two other methods used are based on the energ euation for lateral buckling. The first of these uses hand calculations and a limited series for the buckled shape, while the second uses a finite element computer program based on cubic deformation fields. Both of these produce solutions which agree closel with the finite integral and approimate solutions for the different differential euation derived in this paper, but are markedl different from the solutions for the previousl used euation. It is concluded that the previousl used torsion differential euation is in error. KEYWORDS Beam, buckling, differential euation, distributed load, elasticit, torsion School of Civil Engineering Research Report r96 Page
TABE OF CONTENTS ABSTRACT... KEYWORDS... TABE OF CONTENTS... INTRODUCTION... 5 DERIVATION OF TORSION DIFFERENTIA EQUATION... 5. Calculus of Variations... 5. Euilibrium of the Twisted Beam... 6 SOUTIONS... 6. Solution b Finite Integrals... 6. Finite Element Solutions... 7. Timoshenko s Solutions... 7. Approimate Solutions... 7 CONCUSIONS... 8 5 REFERENCES... 8 6 APPENDIX FINITE INTEGRAS... 9 7 NOTATION... School of Civil Engineering Research Report r96 Page
. INTRODUCTION The differential euation for the variation of the twist rotation along the centroidal ais of a doubl smmetric beam loaded in the YZ principal plane is reported in [] as being d d d EI w GJ V () d d d EI in which E, G are the elastic moduli, I, J, and I w are the second moment of area about the ais, the torsion section constant, and the warping section constant, V and are the internal shear and moment stress resultants, and is the distance below the centroidal ais at which a distributed load acts. For a simpl supported beam under uniforml distributed load (Fig. ) V 8 () Euation omits an epected term of the tpe and includes an unepected term V d /d. It is asserted that Euation is incorrect. The purpose of this paper is to show how the correct torsion differential euation can be derived and to compare its predictions with those of Euation.. DERIVATION OF TORSION DIFFERENTIA EQUATION. Calculus of Variations The calculus of variations can be used to derive the torsion differential euation from the energ euation for lateral buckling. The energ euation for doubl smmetric sections with distributed loads onl is [] U { " ' " " } T EI u GJ EI w u d () in which U T is the total potential, is the length, u is the lateral displacement, and d/d. According to the calculus of variations, the functions u, which make stationar satisf the euations U T F(, u",, ', ") d () d d F F u" d d F d ' d F " (5) This leads to [] School of Civil Engineering Research Report r96 Page 5
( EI ( EI u")" ( )" ")" ( GJ' )' w u" (6) For beams with end twist rotation prevented, the first of Euations 6 can be integrated to EI u" (7) Substituting this into the second of Euations 6 leads to ( EI ")" ( GJ' )' ( / EI ) (8) w This torsion differential euation includes the term missing from Euation and omits the unepected V d / d term. Reference includes an argument for the inclusion of this unepected term based on the assumption that the internal shear V ma be treated as a vertical eternal force that displaces laterall as the beam deflects and twists.. Euilibrium of the Twisted Beam The torsion differential euation can also be obtained b considering the euilibrium of the applied loads in the buckled position shown in Fig.. For overall euilibrium, the H end reactions consist of a vertical force / and a torue ( u ) d / about the fied Z ais. The global reactants of these and the distributed load at a distance from the H end are, V (= ) and Z ( u ) d / ( ( u ) d (9) acting about the fied Z ais, and the torue resultant of these acting about the displaced and rotated ais is u' V u () Z This torue is resisted b the uniform torsion and the warping rigidities, so that w EI ''' GJ' ( u ) d / ( u ) d u' ' u () Differentiating this euation and using = - leads to the uniform beam version of Euation 8.. SOUTIONS. Solution b Finite Integrals Euations and 8 ma be solved numericall b the method of finite integrals [, 5], as eplained in the Appendi. The data used for an eample are E = E5 N/mm, G = 769 N/mm, I = 8E5 mm, J = 5E mm, I w = 6877E8 mm 6, d w = 7. mm, and = 5 mm, in which d w = (I w /I ) is the distance between flange centroids. Finite integral solutions are given in Fig. for the variation of the dimensionless elastic buckling moment / with the dimensionless load distance P / in which is the value of for = and School of Civil Engineering Research Report r96 Page 6
P EI / P ( GJ EI w / ) (). Finite Element Solutions Finite element solutions for the dimensionless elastic buckling moments have been obtained b using the computer program PRFEB [, 6, 7], and are shown in Fig.. The are in ver close agreement with the finite integral solutions of Euation 8, but differ markedl from the finite integral solutions of Euation.. Timoshenko s Solutions Timoshenko [8] determined approimate solutions for simpl supported beams with uniforml distributed load b using sin / a sin / () in the energ euation (Euation ) and minimising with respect to the undetermined parameter a. The solutions shown in Fig. are in ver close agreement with the finite integral solutions of Euation 8, but differ markedl from the finite integral solutions of Euation.. Approimate Solutions Approimate solutions of the torsion euations ma be made b substituting = sin / and integrating each term over the beam length. Thus for Euation, { EI "" GJ" V ' ( / EI ) } d () w leads to P ( ) (5) which can be solved for values of ( / ) for given values of P /. These solutions have been used to determine the values of / shown in Fig.. These values are uite close to the finite integral solutions of Euation shown in Fig. but ver different from the values obtained for Euation 8. ore accurate solutions could be obtained b using Euation and finding the values of the parameter a which minimise the solutions, in much the same wa as did Timoshenko [8] for his energ method solutions. Using the same method for Euation 8 leads to 8 P ( ) (6) The solutions of this have been used to determine the values of / shown in Fig.. These values are reasonabl close to the finite integral solutions of Euation 8 shown in Fig., but ver different from the values obtained for Euation. The reason for this can be seen to be the change School of Civil Engineering Research Report r96 Page 7
of the value of in Euation5 to 8 in Euation6, which suggests that Euation underestimates the significance of the load distance b a factor of.. CONCUSIONS A torsion differential euation previousl used [] for analsing the elastic lateral buckling of simpl supported doubl smmetric beams with distributed loads acting awa from the centroidal ais omits an epected term and includes an unepected term. A different euation which includes the epected term and omits the unepected term is derived in this paper b two different methods. The first method uses the calculus of variations with the second variation of the total potential, while the second method considers the euilibrium of the deflected and twisted beam. A number of different methods are used to find solutions for the elastic buckling of beams with uniforml distributed loads. Two of these solve the torsion differential euations numericall. The first method uses a computer program based on finite integrals [, 5], while the second uses hand calculations with a single term approimation of the buckled shape. These methods produce different solutions for the two torsion differential euations. Two other methods used are based on the energ euation for lateral buckling. The first of these b Timoshenko [8] uses hand calculations and a limited series for the buckled shape, while the second uses a finite element computer program [6, 7] based on cubic deformation fields []. Both of these produce solutions which agree closel with the finite integral and approimate solutions for the different differential euation derived in this paper, but are markedl different from the solutions for the previousl used euation. It is concluded that the previousl used torsion differential euation is in error. 5. REFERENCES [] amb, AW and Eamon, CD (5), oad height and moment factors for doubl smmetric wide flange beams, Journal of Structural Engineering, ASCE, (). [] Trahair, NS (99), Fleural-Torsional Buckling of Structures, E & FN Spon, ondon. [] Baant, ZP and Cedolin, (99), Stabilit of Structures, Oford Universit Press, New York. [] Trahair, NS (968), Elastic stabilit of propped cantilevers, Civ. Engg Trans, Inst. Engrs, Aust., CE(), 9-. [5] Brown, PT and Trahair, NS (968), Finite integral solution of differential euations, Civ. Engg Trans, Inst. Engrs, Aust., CE(), 9-96. [6] Hancock, GJ and Trahair, NS (978), Finite element analsis of the lateral buckling of continuousl restrained beam-columns, Civ. Engg Trans, Inst. of Engrs, Aust., CE(), -7. [7] Papangelis, JP, Trahair, NS, and Hancock, GJ (99), Computer analsis of elastic fleuraltorsional buckling, Journal of the Singapore Structural Steel Societ, (), 59-67. [8] Timoshenko, SP and Gere, J (96), Theor of Elastic Stabilit, cgraw-hill, New York. School of Civil Engineering Research Report r96 Page 8
6. APPENDIX FINITE INTEGRAS In the method of finite integrals [, 5], a differential euation is replaced b a set of simultaneous euations which represent the differential euation at each of a number of points along a beam, as in the method of finite differences. However, the unknowns in these euations are the highest order differential operators at the points, instead of the lowest order as in the method of finite differences. This allows the use of integration which is more accurate than differentiation, leading to faster convergence or more accurate solutions. In addition, the boundar conditions are treated naturall using the constants of integration, and no fictitious points are reuired. The terms of Euation ma be represented b using ' '' " ' "" d A "" dd A A "" ddd A "" dddd A / A A / 6 A / A A (A-) in which A A A A ( ' '') ( ") ( ' ) ( ) (A-) Boundar conditions of () = and ( ) = reuire A = A =. Boundar conditions of () = and ( ) = reuire A A "" "" dddd A dd / 6 (A-) If the beam is divided into an even number n of eual intervals b n+ nodes, then each continuous integral ma be replaced b combinations of the values of the integrand at the nodes, such as ' '' d ( h /)[ N]{ ""} (A-) in which h = /n is the interval length and [N] is the integration matri. 5 8. 6. [N] 6 9 8. (A-5) 6 8 6........ 6 8 6 8 6 School of Civil Engineering Research Report r96 Page 9
which is based on fitting a series of parabolas to the integrand. Thus Euation is replaced b [ T ]{ ""} {} (A-6) in which [ w I T] EI [ I ] GJ[ I ] V [ I ] ( / EI )[ ] (A-7) In this euation, [I ] is a unit matri and [ I [ I [ I ] ( h /) [ N][ N] A [ ] ] ( h /) [ N][ N][ N] A [ ] ( h /) [ N][ N][ N][ N] A [ ]/ A [ I ] ] / 6 A [ ] (A-8) in which [ ], [ ], and [ ] are diagonal matrices with the appropriate values of,, and, and A ( h /) { NN A ( h /) { NNNN T } { ""}/ T } { ""}/ A / 6 (A-9) in which {NN } T and {NNNN } T are the last lines of [N][N] and [N][N][N][N] respectivel. Similar replacements are made for Euation 8. The elastic buckling loads ma be determined b finding the values which satisf T (A-) Because Euation A-6 is uadratic in the load, an iterative process will be reuired for this. School of Civil Engineering Research Report r96 Page
7. NOTATION a Parameter in buckled shape A - Constants of integration d w Web depth E Young s modulus of elasticit F Function G Shear modulus of elasticit h Interval = / n I inor ais second moment of area I w Warping section constant [I - ] Finite integral matrices J Uniform torsion section constant ength Bending moment aimum moment at elastic buckling Uniform bending elastic buckling moment Torue about displaced ais Z oment about fied Z ais Value of for = n Number of intervals [N] Integrating matri {NN} T ast line of [N][N] {NNNN} T ast line of [N][N][N][N] P Column elastic buckling load Intensit of distributed load [T] Total torsional stiffness matri u ateral deflection U T Total potential V Shear, Principal aes Distance of load below centroidal ais Buckled centroidal ais Z Fied centroidal ais [,, ] Diagonal matrices of values of,, Angle of twist rotation School of Civil Engineering Research Report r96 Page
/ / Fig. Simpl Supported Beam / V Z Z du/d u X Fig. Euilibrium of Buckled Beam School of Civil Engineering Research Report r96 Page
Finite integral solutions Euation Euation 8 Finite element Timoshemko [8] = sin /.5. Value of /.5..5 Value of P / Fig. Solutions of Differential Euations School of Civil Engineering Research Report r96 Page