Buckling loads of columns of regular polygon cross-section with constant volume and clamped ends

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1 76 Bukling loads of olumns of regular polygon ross-setion with onstant volume and lamped ends Byoung Koo Lee Dept. of Civil Engineering, Wonkwang University, Iksan, Junuk, 7-79, Korea Sang Jin Oh Dept. of Civil Engineering, Provinial College of Damyang, Damyang, Chonnam, 17-8, Korea Guangfan Li Dept. of Civil Engineering, Yanian University, Yanji 133, China ABSTRACT A numerial method is developed for alulating the ukling loads of tapered olumns of regular polygon ross-setion with onstant volume and oth lamped ends. The linear, paraoli and sinusoidal tapers are onsidered in numerial examples. From the numerial results, the strongest olumns y the taper types and side numers of regular polygon ross-setions are identified. KEYWORDS Bukling load; onstant volume; strongest olumn; tapered olumn 1. Introdution Sine olumns are asi strutural forms, these units have een widely used in various engineering fields. Estimating the ukling loads of non-prismati olumns, whih have the same volume with speifi span length, are attrative in the viewpoint of optimal design. Sine Lagrange [1] had attempted to determine the optimum shape for a olumn, many investigators inluding Keller [], Tadjakhsh and Keller [3], Barnes [] and Cox and Overton [] determined the shape of the strongest olumn. In this study, the strongest olumn is defined as the elasti olumn of given oth length and volume, whih an arry the highest axial load without ukling. Lee and Oh [6] alulated the ukling loads of olumns of onstant volume. Nowhere in the open literature, the solutions for the lass of ukling prolems onsidered herein: ukling loads of non-uniform or tapered olumns of regular polygon ross-setion with onstant volume and oth lamped ends, whose ross-setional depths are varied y funtional fashions are given. The purpose of this paper is to investigate the ukling loads of suh olumns and the onfigurations of strongest olumns.. Ojet olumn Shown in Figure 1(a) is the ojet olumn of speifi span length l and of onstant volume V, whih is supported y oth lamped ends. All the olumns analyzed in this study have the same span length and same volume. Its ross-setional shape is the regular polygon whose ross-setional depth depited as h varies with the axial length s. The area and area moment of inertia of ross setion depited as A and I, respetively, vary with s. Figure 1() shows the

2 77 variation of depth h with s. As shown in this figure, the depth h is varied y funtional fashion, and depths h at s= and l, and at s=l/ are h and h m, respetively. For defining the geometry of olumn, a non-dimensional system parameter or setion ratio n is introdued as follows. n = h h (1) m The quantities A and I of the regular polygon ross-setion with integer m of side numer and ross-setional depth h are expressed in the forms where A = 1 h, () I = h (3) 1 = msin( π m) os( π m), (.1) 3 = msin( π m) os ( π m)[ 1+ tan ( π m) / 3] /. (.) Figure 1 - (a) Column of regular polygon ross-setion with onstant volume and () Its variation of ross-setional depth. Now, onsider the funtional equations of variale depth h. It is natural that all olumns whose variale depths are presried should e the ojet ones. In this study, the linear, paraoli and sinusoidal tapers are hosen for the variale depth h of tapered olumn. First, the equation h of linear taper through three points of (, h ), (l/, nh ) and (l, h ) in retangular o-ordinates (s, h) is otained. The result is h = h [3(s/ l) + 1], s l/, () h = h [ 3(s/ l) ], l/ s l, Where 3 = n 1. (6) The olumn's volume V an now e alulated y using equations () and (): where l V = Ads = ( h l ), (7) 1

3 78 = V (1h l ) = (n + n + 1) 3. (8) In aove equation (8), the oeffiient is defined as a ratio of onstant volume V to volume of uniform olumn of regular polygon ross-setion with depth h, 1 h l. Seond, the equation h and oeffiient of paraoli taper are, respetively, h = h [ 3 (s / l) + 3 (s / l) + 1], s l, (9) = (8 n + n+ 3) 1. (1) Finally, the equation h and oeffiient of sinusoidal taper are, respetively, h = h [3 sin( πs / l) + 1], s l, (11) = ( n 1) + ( n 1) π + 1. (1) In equations (9) and (11), the oeffiient 3 is defined in previous equation (6). 3. Mathematial model The ojet olumn is sujeted to a ompressive load P as shown in Figure. The olumn sujeted to a load P less than the ukling load B is perfetly straight. But when the P exeeds the B, the olumn is ukled. The dashed line and solid urve are the neutral axes of the unukled and ukled olumns, respetively. Thus the shape of elastia is the solid urve defined y the (x, y) o-ordinate system whose origin is at left end. At material point (x, y), the olumn's ar length is s, and the variale area moment of inertia of ross setion taken with respet to s is I disussed in aove setion. Also the rotation of ross-setion and ending moment are depited as θ and M, respetively. It is noted that the axis length of ukled olumn maintains its length l due to inompressiility of olumn, and therefore the value s at right end is l. The end moments at oth ends (s= and s=l) are M. It is assumed that Bernoulli-Euler theory governs the ukled olumn ehavior under load, for whih the differential equations for the elastia are Figure - Variales of elastia of ukled olumn. dθ ds = (M Py) EI, s l, (13) dx ds = os θ, s l, (1)

4 79 dy ds = sin θ, s l, (1) where E is Young's modulus and the term of (M -Py) in equation (13) is the ending moment M at the material point (x, y). Sine the horizontal and vertial displaements and rotation at left end (s=) are not allowed, the following oundary onditions are otained: x = at s =, (16) y = at s =, (17) θ = at s =. (18) Sine the rotation at mid-point of ukled olumn axis (s=l/) is zero due to the symmetry of olumn geometry, the oundary ondition is θ = at s = l/. (19) When the differential equations (13) - (1) with the oundary onditions of equations (16) - (19) are solved y the appropriate numerial methods, the non-linear ehaviors of ukled olumns suh as the elastia and equilirium path are otained. However, suh prolem is eyond the purpose of this study. To failitate the numerial studies and to otain the most general results for this lass of prolem, the axial load, the end moment, the o-ordinates, and the governing differential equations with their oundary onditions are ast in the following non-dimensional forms. The load parameters p and m are defined as p = Pl ( π EIe ), () m = M l ( π EI e ), (1) where I e is the area moment of inertia of irular ross-setion of uniform olumn whose volume is V, defined as I e = V (πl ). () Note that the load parameters p and m of equations () and (1) with equation () are defined y using the onstant volume V and the olumn length l in order to ompare all the responses of olumns regardless of taper type, side numer m and setion ratio n. The ar length s and oordinates (x, y) are normalized y the olumn length l: λ = s l, (3) ξ = x l, () η = y l. () When equation (3) is omined with either equation () or equation (9) or equation (11), and equations ()-() are used, the non-dimensional form of equation (13) eomes

5 8 where ( m pη) ( i), λ 1 dθ dλ = π1, (6.1) i = ( 3λ + 1), λ., for linear taper : (6.) i = ( 3λ ),. λ 1, for paraoli taper : i = ( λ + λ + 1), λ 1, (6.3) 3 3 for sinusoidal taper : i = [ sin( πλ) + 1], λ 1. (6.) 3 It is realled that the oeffiients 1 - of differential equation (6.1) with equations (6.)- (6.) ontain the side numer m and setion ratio n, respetively, as shown in the previous setion. Just after the olumn is ukled, all values of olumn ehavior inluding m are losed to zero. In this study, the ukling load parameter is approximately equivalent to the load parameter p whose end moment m is 1 1-1, i.e. nearly zero ut not zero. Sustituting m =1 1-1 and p= into equation (6.1) gives 1 ( 1 1 η) ( i), λ 1 dθ dλ = π in whih the ukling load parameter is defined as 1 (7.1) = Bl ( EIe ). (7.) Further, with equations (3)-(), equations (1) and (1) eome dξ dλ = osθ, λ 1, (8) dη dλ = sin θ, λ 1. (9) The non-dimensional forms for oundary onditions of equations (16)-(19) are otained y equations (3)-(): ξ = at λ =, (3) η = at λ =, (31) θ = at λ =, (3) θ = at λ = 1. (33). Numerial methods Based on aove analysis, the algorithm was developed to solve differential equations (7.1), (8) and (9) for alulating the ukling load parameter. The Runge-Kutta and Regula-Falsi methods were used to integrate differential equations and to determine the for a given geometry of olumn. This algorithm is summarized as follows. (1) Speify taper type (linear/paraoli/sinusoidal) and geometry (m and n), and alulate 1 -. () Assume a trial value in whih the first trial value is. (3) Integrate equations (7.1), (8) and (9) with the oundary onditions of equations (3)-

6 81 (3) in the range from λ= to 1/ using the Runge-Kutta method. The results give trial solutions for θ=θ(λ), ξ=ξ(λ) and η=η(λ). () Set D=θ (1/). If the value of assumed in step is the harateristi value of the elastia, then D must e zero due to equation (33). The first riterion for onvergene of the solutions is D () If the value of D does not satisfy the first onvergene riterion, then inrement the previous trial value. (6) Repeat steps (3)-() and note the sign of D in eah iteration. If D hanges sign etween two onseutive trial values 1 and, then the harateristi value lies etween 1 and. (7) Compute an improved value 3 ased on its two previous values 1 and using the Regula- Falsi method. The seond riterion for onvergene of solutions is ( - 1 )/ for whih D 1 D < in whih D 1 and D are the D values orresponding 1 and, respetively. (8) Terminate the alulations when two onvergene riteria are met. Print the as the approximate ukling load parameter. Based on this algorithm, a FORTRAN omputer program was written to solve the ukling load. All omputations were arried on a noteook omputer with graphis support. For all of the numerial results presented herein, a step size of λ=(1/)/ in the Runge-Kutta method was found to give onvergene for to within three signifiant figures.. Numerial results and disussions For the purpose of validation of this study, the ukling load parameters predited y the present theory are ompared to those availale in referenes [7,8] in Tale 1. This tale shows the results of this study agree quite well with the referene values, in whih m= in geometry olumn means the irular ross-setion, namely m=. n = n = n = n = Tale 1 - Comparisons of etween this study and referenes Geometry This study Bukling load parameter, Referene n=1. *, m= =.. of Ref. [7].836, m.836, m.836, m.836, m paraoli = 3 = = = = = = = of.1.76 Ref.[8] *If n=1. the olumns are uniform regardless of taper types. See equations (6.)-(6.). Shown in Figures 3- are the versus n urves of olumns with m=3,, and for linear, paraoli and sinusoidal taper, respetively. Eah urve reahes a peak, whih is marked y. At these peak points, the olumns orresponding to the given taper types have the largest values, whih are the ukling load parameters of strongest olumns. Here the word strongest

7 8 is used to mean most resistant to ukle. It is found that all strongest olumns our at the same value n regardless of side numer m if the taper type is same. And all values of strongest olumns derease, as the value m is inreased from 3 to to to. The values of and n of all strongest olumns are summarized in tale. From this tale, it is noted that all values of strongest olumns are largest at m=3 (triangular ross-setion) and smallest at m= (irular ross-setion), and the ratios of of m=3 to of m= are same, i.e. 1.1, regardless of taper types. Also, this holds true that eah ratio of of m= and to of m= is same regardless of taper types linear taper : strongest olumns m= paraoli taper : strongest olumns m= n. 1 3 n Figure 3 - vs. n urves of linear taper Figure - vs. n urves of paraoli y side numer m. taper y side numer m sinusoidal taper : strongest olumns m= n Figure - vs. n urves of sinusoidal taper y side numer m. Shown in Figure 6 are the versus n urves of paraoli, sinusoidal and linear tapers, respetively, for m=3, in whih the strongest olumns are marked y. It is lear that the strongest of all olumns y taper type is the paraoli tapered olumn as shown in this figure and Tale. The effet of taper type on is negligile when n is less than aout..

8 triangular ross-setion(m=3) : strongest olumns. 3. L S P. 1. P : paraoli taper S : sinusoidal taper L : linear taper. 1 3 n Figure 6 - vs. n urves y taper type. Tale - Values of n and of strongest olumns y taper type and side numer m Taper type m n Ratio* Linear taper 3 Paraoli taper 3 Sinusoidal taper * Ratio of of m=3, and, respetively, to of m=. 6. Conluding remarks A novel numerial method developed herein for omputing the ukling load of tapered olumn of regular polygon ross-setion with onstant volume and oth lamped ends was found to e effiient, and highly versatile. The linear, paraoli and sinusoidal tapers were hosen for the variale ross-setional depth. As the numerial results, the ukling load parameters versus setion ratio ( vs. n) urves were reported. The strongest olumns y taper types and side numers of regular polygon ross-setion were identified y reading the peak points of ukling load parameters and their orresponding setion ratios on versus n urves. ACKNOWLEDGEMENT This work was supported y Wonkwang University in 1. The first author aknowledges this finanial support.

9 8 REFERENCES 1. Todhunter, I. and Pearson, K., A History of the Theory of Elastiity and of the Strength of Materials, Camridge Press, Keller, J. B., The shape of the strongest olumn, Arhive for Rational Mehanis and Analysis, Vol., 196, pp Tadjakhsh, I. and Keller, J. B., Strongest olumns and isoperimetri inequalities for eigenvalues, Journal of Applied mehanis, ASME, Vol. 9, 196, pp Barnes, D. C., The shape of the strongest olumn is aritrarily lose to the shape of the weakest olumn, Quarterly of Applied Mathematis, Vol. 6, 1988, pp Cox, S. J. and Overton, M. I., On the optimal design of olumns against Bukling, SIAM Journal on Mathematial Analysis, Vol. 3, 199, pp Lee, B.K. and S.J. Oh, Elastia and ukling load of simple tapered olumn with onstant volume, International Journal of Solids and Strutures, Vol. 37, No. 18, pp Timoshenko, S.P. and Gere, J. M., Theory of Elasti Staility, MGraw-Hill, Lee, B. K. and Mo, J. M., Free virations and ukling loads of tapered eam-olumns of regular polygon ross-setion with onstant volume, Journal of Korean Soiety of Noise and Viration Engineering, Vol. 6, No., 1996, pp

Lecture 11 Buckling of Plates and Sections

Lecture 11 Buckling of Plates and Sections Leture Bukling of lates and Setions rolem -: A simpl-supported retangular plate is sujeted to a uniaxial ompressive load N, as shown in the sketh elow. a 6 N N a) Calulate and ompare ukling oeffiients