Int. J. Engng Ed. Vol. 14, No. 3, p. 204±216, 1998 0949-149X/91 $3.00+0.00 Printed in Great Britain. # 1998 TEMPUS Publications.

Interesting Instructional Problems in Column Buckling for the Strength of Materials and Mechanics of Solids Courses*

J. NEURINGER and I. ELISHAKOFF Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, FL 33431-0991, USA. E-mail: [email protected] In this study the following problem is addressed: a uniform column is subjected to a compressive load; an additional support is placed to increase the buckling load. The following question is posed: where to place the support location so as to maximize and evaluate the resulting buckling load? It turns out that this question can be effectively dealt with in the standard courses of Strength of Materials, Mechanics of Solids, or Mechanics of Materials, since all the necessary tools needed are presently uniformly taught in these existing courses. Including this interesting case into the curriculum may enhance students' grasp of the subject, sharpen their mind, and trigger an additional interest in the exciting subject of theoretical and applied mechanics. Topics covered in sections 1±3 can be taught in one or two 50-minute lectures, whereas the exposition of the entire material may take between two to three 50-minute lectures, depending on the interest of students.

SUMMARY OF EDUCATIONAL ASPECTS report results on optimization of columns, plates OF THE PAPER and shells under buckling conditions [11, 12]. It appeared to the authors, that the motivation of 1. The paper discusses materials for a course in the students can be enhanced, if some model Strength of Materials and Mechanics of Solids. problems can be included on more advanced 2. Students of second-year Mechanical Engineer- topics in the undergraduate courses. This would ing, Ocean Engineering, Civil Engineering and provide students with some `raisins to look for'. As Aerospace Engineering are taught this course. Budiansky and Hutchinson [13] note, `Everyone 3. The mode of presentation is by lecture and is loves a buckling problem'. We propose some run as a regular course. problems the solutions to which hopefully will 4. Hours required to cover the material is 4 to 5 enlarge the number of `buckling lovers,' to include with 2 to 3 revision hours. more undergraduate students. Maybe even every 5. The novel aspects presented in this paper hope- undergraduate textbook should include a section fully help to sharpen the minds of the students on `Some Interesting Problems' providing, gastro- and increase interest in applied mechanics. nomically speaking, the dessert after the main 6. The standard text recommended for the course serving of essential material is taught. We deal is Hibbeler, Mechanics of Materials [2]. here with the problem of locating the intermediate support in a uniform column with ideal boundary conditions so as to maximize the buckling load. The analysis is conducted via the straightforward 1. INTRODUCTION optimization. It turns out that with the knowledge acquired by the undergraduate students in the NATURALLY THERE IS a big gap between standard course, some interesting problems can the exposition of the topic of buckling in the be solved, even those with seemingly un-intuitive undergraduate textbooks devoted to Mechanics solutions. of Solids, Strength of Materials or Mechanics of Note that the beams and columns with intermedi- Materials [1±5] and more specialized monographs, ate supports have been dealt with by Olhoff and suitable for the elective or graduate courses [6±10]. Taylor [14], Rozvany and Mroz [15], Wang, et al. The undergraduate textbooks invariably report [16±19], Liu, et al. [20, 21], and Hou [22] in various buckling of uniform columns with various bound- optimization contexts. The above studies used ary conditions, whereas the advanced texts include Bernoulli-Euler theory. Ari-Gur and Elishakoff numerous results of uniform or non-uniform [23] studied the effect of shear deformation on columns, plates and shells, pertinent to engineering buckling of columns with intermediate support. practice. Among other topics, specialized books Here we deal with the instructional side of this problem for uniform columns with different * Accepted 1 November 1997. boundary conditions.

204 Interesting Instructional Problems in Column Buckling 205

2. UNIFORM COLUMN SIMPLY- The continuity conditions read: SUPPORTED AT BOTH ENDS, WITH AN INTERMEDIATE SUPPORT w1 a† ˆ w2 a† ˆ 0 8† w 0 a† ˆ w 0 a† 9† The buckling of a uniform column is governed 1 2 by the following differential equation: 00 00 w1 a† ˆ w2 a† 10† d 4w d2w Equation (9) signifies the continuity of the slopes, EI ‡ P ˆ 0 1† dx4 dx2 whereas equation (10) indicates the continuity of bending moments. where w x† is the transverse displacement, The general solution w1 x† in the first region is: x ˆ axial coordinate, E ˆ modulus of elasticity, w x† ˆ A sin kx ‡ A cos kx ‡ A x ‡ A 11† I ˆ moment of inertia, P ˆ compressive loading. 1 1 2 3 4 We divide equation (1) by flexural stiffness EI, and Satisfying the conditions of equation (6) and the denote: first of the conditions in equation (8) we get:   P 2 x ˆ k 2† w x† ˆ A sin kx sin ka 12† EI 1 1 a to obtain: The general solution w2 x† in the second region is: 4 2 d w 2 d w ‡ k ˆ 0 3† w2 x† ˆ B1 sin kx ‡ B2 cos kx ‡ B3 x ‡ B4 13† dx4 dx2 where j's are the constants of integration. After At the cross-section x ˆ a, an additional support is satisfaction of the conditions in equation (7) and placed (Fig. 1a). Thus, two regions are created. the second condition, equation (8), equation (13) For the purpose of identification the displacement reduces to: in the first region is denoted by w1 x†, whereas the   displacement in the second region a < x  L†, is L x w2 x† ˆ B1 sin k L x† sin k L a† denoted by w2 x†. Thus, instead of equation (3) we L a have the following two equations: 14† 4 2 Equation (9) results in: d w1 2 d w1 ‡ k ˆ 0; for 0  x < a 4†   dx4 dx2 sin ka A k cos ka 4 2 1 a d w2 2 d w2 ‡ k ˆ 0; for a  x  L 5†   dx4 dx2 sin k L a† ‡ B1 k cos k L a† ˆ 0 The boundary conditions read: L a 15† w 0† ˆ w 00 0† ˆ 0 6† 1 1 Equation (10) yields: 00 w2 L† ˆ w2 L† ˆ 0 7† A1 sin ka B1 sin k L a† ˆ 0 16†

Non-triviality of A1 and B1 yields the transcen- dental equation:   sin k L a† sin ka k cos k L a† L a   1 ‡ sin k L a† k cos ka sin ka ˆ 0 a 17† Introducing non-dimensional quantities: a u ˆ ; ˆ kL; 18† L and using trigonometric identities we arrive at the following characteristic equation: Fig. 1. (a) Uniform column, simply supported at its ends 1 with an intermediate support. (b) Optimal location of an 2 sin ‡ ‰cos cos 2u 1†Š ˆ 0 19† intermediate support is in column's middle. u u2 206 J. Neuringer and I. Elishakoff

Now we turn to determining a location a ˆ aà of We conclude that the best location for the the intermediate support, such that the buckling intermediate support, if the column is simply load will attain a maximum at a ˆ aà corresponds supported at both ends is the middle cross section to u ˆ aÃ=L ˆ uÃ. We first note that one of the (Fig. 1b), with the buckling load given in equation arguments on the left hand side of equation (19), (29). Let us consider a limiting case. denoted by f: f u†; u† ˆ 2 sin 3. BUCKLING LOAD OF A SIMPLY- 1 ‡ ‰cos cos 2u 1†Š 20† SUPPORTED COLUMN WITH u u2 INTERMEDIATE SUPPORT'S LOCATION APPROACHING ZERO namely u†, is an implicit function of u. We wish to determine the derivative d =du and set it equal Taking into account the expressions for the to zero. We proceed as follows. Differentiating transverse displacement w1 x† and w2 x† given in equation (19) with respect to u, we get: equations (12) and (14), respectively, we obtain the @ f d @ f following formulas for the slopes at the intermediate ‡ ˆ 0 21† support location: @ du @u Thus,      0 sin ka d @ f @ f w a† ˆ A1 k cos ka 30† ˆ 22† 1 a du @u @   1 On the assumption that @ f =@ differs from zero, w 0 a† ˆ B k cos k L a† sin k L a† 2 1 L a we see that the condition: 31† d ˆ 0 23† du Now, when a approaches zero (Fig. 2a), we get: is equivalent to: @ f   ˆ 0 24† 0 1 lim w1 a† ˆ A1 lim k cos ka sin ka @ u a ! 0 a ! 0 a The latter equation yields:   1 ˆ A1 k lim cos ka lim sin ka 1 2u a ! 0 a ! 0 a ‰cos cos 2u 1†Š u u2†2 ˆ A1 k k† ˆ 0 32† 2 ‡ sin 2u 1† ˆ 0 25† u u2† 0 0 Since condition (9) requires w1 a† ˆ w2 a† for all a, we thus also require that: We are interested in the smallest non-zero solution for of the coupled set of equations (19) and (25), 0 and call it cr; the corresponding value of u is lim w2 a† ˆ 0 33† a ! 0 called ucr. Let us examine equation (25). It is satisfied by uà ˆ aÃ=L ˆ 1=2 independent of the value of . Hence, by putting u ˆ uà ˆ 1=2 into equation (19) we get:   sin cos 2 sin ˆ 0 26† 2 2 2 Equation (26) yields two equations. Either:

sin ˆ 0 with ˆ 2 27† 2 or,

tan ˆ with ' 8:986 28† 2 2 The smaller value is ˆ 2, resulting in the buckling load: Fig. 2. (a) A simply-supported column with intermediate support's location approaching zero. (b) Within the Bernoulli- EI 42EI Euler theory, when an intermediate support approaches the 2 left end, the boundary condition there tends to that of the Pcr ˆ cr ˆ 29† L2 L2 clamped end. Interesting Instructional Problems in Column Buckling 207

Thus, The boundary conditions read, in the new circumstances: 0 lim w2 a† a ! 0 0   w1 0† ˆ w1 0† ˆ 0 36† 1 0 ˆ B1 lim k cos k L a† sin k L a† w2 L† ˆ w L† ˆ 0 37† a ! 0 La 2   1 whereas the continuity conditions remain ˆ B1 k cos kL sin kL ˆ 0 34† unchanged, and are given by equations (8±10). In L the first region the solution that satisfies the or, boundary conditions and the first of the equation (8) reads (see Appendix A): tan kL ˆ kL 35† w x† ˆ A‰ sin k x a† sin ka ka cos kx which is the characteristic equation of the column 1 that is clamped at the left end and simply ‡ kx cos ka ‡ sin kx k x a†Š 38† supported at the right end. Thus, within the Bernoulli-Euler theory, when In the second region the transverse displacement the intermediate support approaches the left end, becomes (see Appendix B): the boundary condition at the left support tends to that of the clamped end (Fig. 2b). Note that to w2 x† ˆ Cfk L a† cos k x L† ‡ sin k x L† the students who will be surprised with this con- sin k x a† ‡ sin k L a† clusion, the lecturer should explain that within the more refined Timoshenko beam theory this kL cos k L a† ‡ ak `phenomenon' does not take place: when two supports tend to each other, they again make the ‡ ‰k cos k L a† kŠxg 39† simply support condition, as expected. We observe that a more difficult theory leads to the result that where A and C are arbitrary constants. The two can be `seen' immediately, whereas the simplified remaining continuity conditions in equations (9) theory of Bernoulli-Euler leads to a less `digestible' and (10) yield: result. This problem is addressed by Ari-Gur and 2 Elishakoff [23]. A 2k cos ka ‡ k a sin ka 2k† ‡ C‰k2 L a† sin k a L† 4. UNIFORM COLUMN CLAMPED AT 2k cos k a L† ‡ 2kŠ ˆ 0 40† BOTH ENDS, WITH AN INTERMEDIATE SUPPORT k3a cos ka k2 sin ka†A

Consider now a uniform column that is ‡ ‰k3 L a† cos k a L† clamped at both ends and is supported by an additional support at the location x = a (Fig. 3a). ‡ k2 sin k a L†ŠC ˆ 0 41† We again have two regions, with attendant govern- ing differential equations as per equations (4) and Setting the determinant of the system (40) and (41) (5). equal to zero results in the characteristic equation: 2 cos ka ‡ ka sin ka 2†  ‰k L a† cos k a L† ‡ sin k a L†Š ‡ sin ka ka cos ka†  ‰k L a† sin k a L† 2 cos k a L† ‡ 2Š ˆ 0 42†

Using again the notations as per equation (18) transforms the characteristic equation into:

2 cos u ‡ u sin u 2†  ‰ 1 u† cos 1 u† sin 1 u†Š ‡ sin u u cos u†

Fig. 3. (a) Uniform column clamped at its ends with an  ‰ 1 u† sin 1 u† intermediate support. (b) Optimal location of an intermediate support is in column's middle. 2 cos 1 u† ‡ 2Š ˆ 0 43† 208 J. Neuringer and I. Elishakoff

Using trigonometric identities simplifies this claiming that (1) we have not checked all possible equation into: boundary conditions, (2) that presumably for this theorem to be correct one should mention the f u†; u† ˆ ‰ 2u 1 u† 2Š sin symmetry in boundary conditions and change the statement into: `In order to maximize the buckling ‡ cos ‡ cos u cos 1 u† load of a uniform column with symmetric bound- 2 1 u† cos 1 u† ary conditions (i.e. identical boundary conditions) at its ends, an intermediate support should be ‡ 2 sin 1 u† ‡ 2 sin u placed in the middle cross section.' 2 u cos u ˆ 0 44† If the class is sufficiently strong, the lecturer may proceed to discuss additional cases, which will Setting the partial derivative of the left-hand side reveal that there are more fundamental considera- of equation (44) with respect to u equal to zero tions pertinent to the buckling of a column with yields: intermediate support, than those associated with 2 2 the symmetry. The lack of symmetry in the bound- 1 2u† sin ‡ sin 1 2u† ary conditions of the column simply supported at 2 2 1 u† sin 1 u† one end and clamped at the other should appear sufficient to decide that the intermediate support ‡ 2 2u sin u ˆ 0 45† should be placed elsewhere (not in the middle cross-section) in order to maximize the buckling It is immediately recognized that this equation is load. satisfied by uà ˆ 1=2. Substituting uà ˆ 1=2 into We also observe that the location of the addi- the equation (44) we obtain: tional support that makes the buckling load maxi- mum corresponds to the location of the node in the ˆ 4 46† second buckling node of the columns considered in the above two cases. The interest of the second That is, exactly as in the column that is simply author in this problem arose when he attended the supported at both ends, the location of the inter- International Conference on Vibration Engineer- mediate support that maximizes the buckling load ing in Beijing, People's Republic of China, in 1994, is in the middle of the column (Fig. 3b). The and sat next to Prof. H. C. Hu. The presentation of appropriate buckling load equals: Ref. 20 was made by Dr. Z. S. Liu. During the 162EI presentation, which dealt with the vibration prob- P ˆ 47† cr 2 lem of the beam, Dr. Liu formulated the following L theorem: Can we draw a general conclusion from consid- The location of the additional support to maximize ering these two problems together? It appears that the natural frequency of the beam coincides with that this question should be addressed in the classroom. of the node of the second vibration mode of the beam Some students may suggest that due to the uni- without additional support. formity of the column and the symmetry of the The second author asked Professor Hu if one boundary conditions with respect to the middle could foresee this interesting theorem. The reply cross-section, either the maximum or the minimum was affirmative: `Yes, I have dreamt it!' buckling load should occur when the intermediate One may therefore make a conjecture that for support is placed in the middle cross-section. At the column an analogous theorem would hold: this point the lecturer may point out that in the case of the column without intermediate support The location of the additional support to maximize that is simply supported at its ends, the buckling the buckling load of the column coincides with that of 2 2 the node of the second buckling mode of the column load equals Pcr ˆ  EI=L , whereas placement of the support in the middle cross-section changes the without support. 2 2 buckling load to Pcr ˆ 4 EI=L , i.e. increases it Let us check if this conjecture is satisfied for the four-fold. This suggests that students' anticipa- column that is simply supported at one end and tion that the placement of the additional support clamped at the other, with an intermediate support. in the middle cross-section makes for the extremes of buckling load is correct. The calculation above, however, shows us that this extremum is 5. UNIFORM COLUMN SIMPLY a maximum. SUPPORTED AT ONE END AND CLAMPED The lecturer may ask the students to provide AT THE OTHER END, WITH analogous reasoning for the column that is INTERMEDIATE SUPPORT clamped at the ends. At this juncture the lecturer may ask if the following theorem would be correct: The boundary conditions read (Fig. 4a): `In order to maximize the buckling load of a 00 uniform column, an intermediate support has to w1 0† ˆ w1 ˆ w1 a† ˆ 0 48† be placed in the middle cross section.' We hope w L† ˆ w 0 L† ˆ w a† ˆ 0 49† that some students will refute this statement 2 2 2 Interesting Instructional Problems in Column Buckling 209

In the first region 0  x  a, we have: which reduces to:  x  w x† ˆ A sin kx sin ka 50† 2 1 a ak!1 ‡ a!3 sin ka ‡ !1 sin ‡ !2 sin cos ˆ 0 58† In region 2, a  x  L, we adopt the solution used for the column that is clamped at both ends: Substituting the expressions for !j we get: w2 x† ˆ C‰!1 sin kx ‡ !2 cos kx ‡ !3x ‡ !4Š 51† f u†; u† ˆ sin u cos u u† where  cos u cos ‡ sin u sin † !1 ˆ L a†k sin kL ‡ cos kL cos ka ‡ u sin u cos cos u ‡ sin sin u 1† !2 ˆ L a†k cos kL ‡ sin ka sin kL 52† 2 !3 ˆ k‰cos k L a† 1Š ‡ sin u sin u sin

!4 ˆ sin k L a† kL cos k L a† ‡ ak ‡ cos cos u† ˆ 0 59† The continuity of bending moment: Before proceeding with the analysis of this 00 00 w1 a† ˆ w2 a† 53† equation we first concentrate on investigating the limiting behavior when a ! 0. Let us consider an leads to: expression for the slope in the second region, in the

A sin ka ‡ C !1 sin ka ‡ !2 cos ka† ˆ 0 54† cross-section x ˆ a:

The imposition of the continuity of the slope: 0 w2 a† ˆ C k!1 cos ka k!2 sin ka ‡ !3† w 0 a† ˆ w 0 a† 55† 1 2 ˆ Cfk cos ka‰ L a†k sin kL ‡ cos kL cos kaŠ results in:   kw2 sin ka ‡ k‰cos k L a† 1Šg 60† sin ka A k cos ka a In the limit, when a approaches zero, we get: C k! cos ka k! sin ka ‡ ! † ˆ 0 56† 1 2 3 Lk2 sin kL ‡ k cos kL k ‡ k cos kL k ˆ 0 2 2 Requirement of non-triviality of A ‡ C in equa- 61† tions (54) and (56) results in the characteristic equation: or,

sin ka k!1 cos ka k!2 sin ka ‡ !3† Lk sin kL ‡ 2 cos kL 2 ˆ 0 62†   sin ka k cos ka Using trigonometric identities: a kL kL  ! sin ka ‡ ! cos ka† ˆ 0 57† sin kL ˆ 2 sin cos 1 2 2 2 63† kL 2 cos kL 2 ˆ 4 sin2 2 we get,

kL kL kL 2Lk sin cos 4 sin2 ˆ 0 64† 2 2 2

or, simply,

kL kL tan ˆ 65† 2 2 which in the characteristic equation for the clamped-clamped column. Thus, when a tends to zero, the simply supported end at x ˆ 0 and the Fig. 4. (a) Uniform column simply supported at one end and intermediate support at x ˆ a, tend to act in clamped at the other end, with intermediate support. (b) Optimal placement of an intermediate support is the location concert as a clamped end. of the node in the second bucking mode of a simply supported- We return now to the characteristic equation clamped column without an intermediate support. (59). Recalling the notation of non-dimensional 210 J. Neuringer and I. Elishakoff quantities u and , as per equation (18), the with respect to which the column exhibits sym- characteristic equation is re-written as: metry. Yet, this location is also the node of the second buckling mode of the column without the f u†; u† ˆ uu2† cos ‡2 u sin u u sin intermediate support. Thus, all three cases appear to support the assertion that the buckling load ‡ sin u cos u sin cos † takes a maximum value when the additional sin ‡ cos † sin2 u ˆ 0 66† support is placed at the node of the second buck- ling mode of the column without the intermediate Let us check now the assertion, that the buckling support. Yet, it appears still premature to claim load attains a maximum when the additional that we have hit upon a right conclusion until the support is located at the node of the second two remaining boundary condition cases are buckling node. The characteristic equation for examined. These are the columns involving free the buckling load of the column simply supported ends. at one end and clamped at the other, and lacking an intermediate support, reads: 6. UNIFORM COLUMN SIMPLY tan ˆ 67† SUPPORTED AT ONE END AND FREE AT THE OTHER END, WITH INTERMEDIATE The second root of this equation is: SUPPORT

ˆ 7:7252 68† The expression for the displacement in the first region, satisfying the conditions (Fig. 5a) Substituting equation (67) into equation (66 ), we get: 00 w1 0† ˆ w1 0† ˆ w1 a† ˆ 0 74† 2 sin ‡ cos † sin u 2 u sin u reads: ‡ 2u2 cos ˆ 0 69†  x  w x† ˆ A sin kx sin ka 75† 1 a Solving quadratic equation (69) with respect to sin u, results in: The displacement in the second region, satisfying the conditions: sin u ˆ 00 000 2 0 p w2 a† ˆ w L† ˆ w L† ‡ k w L† ˆ 0 76† 2 u Æ 4 2u2 4 2u2 cos sin ‡ cos † 2 ˆ 2 sin ‡ cos † is: w x† ˆ C‰sin k x L† sin k a L†Š 77† 70† 2 Now, let us calculate the derivatives: Due to equation (67) sin ˆ cos , equation (70)   reduces to: 1 w 0 x† ˆ A k cos kx sin ka u 1 a sin u ˆ 71† 2 ‡ 1† cos 00 2 w1 x† ˆ Ak sin kx 78† Substitution of the numerical value given in w 0 x† ˆ Ck cos k x L† equation (68) yields: 2 00 2 w2 x† ˆ Ck sin k x L† sin 7:725u ˆ 0:9917u 72†

The solution of equation (72) reads:

a ˆ 0:3595L 73† which is the location of the node in the second buckling mode of the simply supported-clamped column without an intermediate support (Fig. 4b). Having arrived at this result one can try to combine three cases of uniform columns con- sidered above, namely (1) the column simply supported at its two ends, (2) the column clamped at its both ends, (3) the column simply supported at one end and clamped at the other. In the first Fig. 5. (a) Uniform column simply supported at one end and two cases the maximum buckling load occurred if free at the other, with intermediate support. (b) In order to the additional support was placed in the middle maximize the buckling load one should place an additional cross-section of the column. This is a cross-section support at column's free end. Interesting Instructional Problems in Column Buckling 211

Imposing the continuity conditions (9) and (10) Bearing in mind equation (88), we get: yields: u ˆ Æm 92†   1 A k cos ka sin ka Ck cos k a L† ˆ 0 where m is zero or an integer. On the other hand a we observe, that through the definition of u in 79† equation (18), it must lie in the closed interval [0, 1]. There are two solutions of equation (92) con- Ak2 sin ka ‡ Ck2 sin k a L† ˆ 0 sistent with this requirement: u = 0 corresponds to 80† m ˆ 0, and u ˆ 1 corresponding to m ˆ 1. The condition u ˆ 0 corresponds to the buckling load The following characteristic equation is obtained, of the clamped-free column without an inter- from equations (79) and (80): mediate support with buckling load:

ak cos ka sin ka† sin k L a† 2EI Pcr ˆ 93† ‡ ak sin ka cos k L a† ˆ 0 81† 4L2 The condition u ˆ 1 corresponds to the buckling or with the familiar non-dimensional variables load of the column that is simply supported at its u ˆ a=L, ˆ kL, we get: both ends, namely, f u†; u† ˆ sin u u cos u† sin u 1† 2EI Pcr ˆ 94† ‡ u sin u cos u 1† ˆ 0 82† L2 Thus, in order to maximize the buckling load of a Using the sum angle formula of trigonometry, simply supported-free column, one should place an equation (82) is reduced to: additional support at column's free end (Fig. 5b). Does the end of the free column correspond to f u†; u† ˆ u sin ‡ sin u sin u 1† ˆ 0 the node of the second buckling mode of the 83† simply supported free column without an addi- tional support? In order to answer this question Calculating the derivative of the left-side of let us derive an expression for the buckling mode equation (83) with respect to u, yields, of such a column. The general solution for the displacement of the uniform column reads: sin ‡ cos u sin u 1† w x† ˆ B sin kx ‡ B cos kx ‡ B x ‡ B 95† ‡ sin u cos u 1† ˆ 0 84† 1 2 3 4 Satisfying the boundary condition w 00 0† ˆ 0 we or conclude that B2 ˆ 0. The condition w 0† ˆ 0 sin ˆ sin 1 2u† 85† leads to B4 ˆ 0. Thus the displacement becomes:

Solving for u in the latter equation we conclude w x† ˆ B1 sin kx ‡ B3 x 96† that: Requirement w 00 L† ˆ 0 results in: 1 2u† ˆ Æ 2m 86† B k2 sin kL ˆ 0 97† where m is zero or an integer: 1 From (97), since B1 6ˆ 0, we require: m ˆ 0; Æ1; Æ2; . . . 87† m Thus k ˆ ; m ˆ 0; Æ1; Æ2; . . . 98† L m u ˆ 88† 000 2 0 Using (96), the condition w L† ‡ k w L† ˆ 0, yields: Substituting this value into the characteristic 3 2 equation (83): B1k cos kL ‡ cos kL† ‡ B3 k ˆ 0  m  Æm sin ‡ sin m† sin Æ 1 ˆ 0 89† Hence, B3 ˆ 0. Thus, the normal modes are given by: or m Wm x† ˆ B1 sin x 99† sin ˆ 0 90† L The smallest non-trivial solution of equation (90) which is the same as those of the column that is is: simply supported at both ends without inter- mediate support. The node of the second buck- ˆ  91† ling mode (m ˆ 2) occurs at L=2. Since we have 212 J. Neuringer and I. Elishakoff demonstrated that the additional support which Requiring non-triviality of A2 ‡ C 2 we arrive at maximizes the buckling load should be placed at the characteristic equation: x ˆ L, we conclude that the conjecture that the buckling load assumes a maximum value when sin k a L†‰ sin ka sin ka ka† cos ka 1†2Š the additional support is placed at the node of the second buckling mode of the column without ‡ cos k a L†‰ cos ka sin ka ka† intermediate support, is violated in the simple ‡ sin ka cos ka 1†Š ˆ 0 106† support-free case. Let us check the remaining case to be able to reach the final conclusions. Using some trigonometric identities and introducing non-dimensional quantities and u, as per equation (18), equation (106) is reduced 7. UNIFORM COLUMN CLAMPED AT ONE to: END AND FREE AT THE OTHER END, WITH INTERMEDIATE SUPPORT f u†; u† ˆ 2 ‡ cos u† sin u 1†

In the first region the column's displacement sin ‡ u cos ˆ 0 107† satisfying the boundary conditions (Fig. 6a): Taking a partial derivative of the left-hand side 0 of equation (107) with respect to u yields: w1 0† ˆ w1 0† ˆ w1 a† ˆ 0 100† reads sin u sin u 1† ‡ 2 ‡ cos ku† cos k u 1†

w1 x† ˆ A‰ cos kx 1† sin ka ka† ‡ cos ˆ 0 108† cos ka 1† sin kx kx†Š 101† Let us set temporarily:

In the second region the solution satisfying the y ˆ u 1† 109† conditions: so that equation (108) can be re-written as follows: w 00 L† ˆ w 000 L† ‡ k2w L† ˆ w a† ˆ 0 102† 2 2 2 sin y ‡ † sin y ‡ ‰2 ‡ cos y ‡ †Š cos y reads ‡ cos ˆ 0 110† w x† ˆ C‰sin k x L† sin k a L†Š 103† 2 Expanding, using the trigonometric formulas for 0 functions of sums and differences of two angles Satisfying the continuity conditions w1 a† ˆ w 0 a†; w 00 a† ˆ w 00 a† yields: and collating like terms results in a quadratic 2 1 2 equation for cosy: A‰ sin ka† sin ka ka† cos ka 1†2Š cos2 y 2 cos cos y ‡ cos2 ˆ 0 111† C cos k a L† ˆ 0 104† which could be rewritten as: A‰ cos ka† sin ka ka† ‡ cos ka 1† sin kaŠ cos y cos †2 ˆ 0 112† ‡ C sin k a L† ˆ 0 105† or, simply, cos y ˆ cos 113† Hence, on the one hand, y ˆ ‡ 2m; m ˆ 0; Æ1; Æ2; . . . 114† and on the other, in accordance to the definition in equation (109): y ˆ u 1† 115† Equating right sides of equations (114) and (115) we arrive at:  m u ˆ 2 1 ‡ 116† 2 Since we require: Fig. 6. (a) Uniform column clamped at one end and free at the other end, with intermediate support. (b) In order to maximize 0  u  1 117† the buckling load, one should place an additional support at column's free end. we immediately observe from equation (116) that u Interesting Instructional Problems in Column Buckling 213 exceeds unity for all non-negative values of m. Let 8. CONCLUSION us, therefore, concentrate on the negative values of m. From equation (116): As is clearly seen in the two last cases the conjecture made by us that the buckling load u ˆ 2 ‡ 2m 118† takes a maximum value when the additional support is placed at the node of the second buck- We substitute this expression into the characteristic ling mode of the column without the intermediate equation (107): support, does not hold. In the last two cases the additional support must be placed at the free end 3 sin ‡ cos 2 sin ‡ 2 ‡ 2m† cos ˆ 0 of the column. Is it possible to put all obtained results as a single statement? The answer is affir- 119† mative. In order to maximize the buckling load of a uniform column by introducing a single support, The solution of this equation is: it must be placed at the location where the first buckling mode of the column without inter- ˆ jmj 120† mediate support achieves a maximum. Indeed, if the displacement at both ends of the column Note that, the absolute value in equation (120) vanishes, then the additional support must be appears since the values of m under consideration placed at the location where the node of the are negative. Substituting this value into equation second buckling mode of the column with the (118) we obtain u ˆ 0. intermediate support removed is located. But this We conclude from the above analysis that there is also a location of the maximum displacement of is no relative maximum or minimum in the open the first buckling mode. interval x 2 0; 1†. However, since the function f, When one of the ends of the column is free, the identified with the left side of equation (107) is additional support must be placed at the free end continuous on a closed interval ‰0; 1Š, it must of the column. This is also a location of the obtain both a maximum value and a minimum maximum value attained by the first buckling value on ‰0; 1Š. A simple check shows that the mode of the column without support. minimum occurs when u ˆ 0 and the maximum Thus all the cases fall in the same, unified is attained when u ˆ 1. category: Substituting this value into equation (107), we In order to maximize the buckling load of a uniform obtain: column with classical boundary conditions at the ends, one first should locate the cross-section where sin ‡ cos ˆ 0 121† the first buckling mode attains a maximum value; one then should place the additional support in that or location. tan ˆ 122† In the actual course the conducted discussions on this topic were lively. We hope that analogous which is the equation associated with the clamped- interest arise in many classes, when teaching the simply supported column. strength of materials course. The course should not Carrying out the corresponding calculation as be reduced to listing of results concisely and was done in the simply supported free case with- quickly covering them; material will be better out additional support, the modes for the clamped- digested if students participate in the discussions free case without additional support are given and are partially elevated to the status of co- by: uncoverers, maybe of even small exciting points.    AcknowledgmentÐ This work has been performed when the 2m 1† x first author studied the course `Elastic Stability' delivered by the wm x† ˆ B1 1 cos  123† second author, within the Lifelong Learning Society (LLS) at 2 L Florida Atlantic University (Director: Ely Meyerson). LLS and the Open University at FAU are attended by over 14,000 senior It is seen that there are no nodes in the second students of South Florida. The help of these organizations in buckling mode (m ˆ 2) in the interval 0; L† note conducting this `engineering teaching enhancement project' is kindly acknowledged. We also appreciate extensive discussions that the third buckling mode (m ˆ 3) has a node at contributed by the attendees of the course, and especially by x ˆ 4L=5. Piero Colajani and Nicola Impollonia.

REFERENCES

1. J. M. Gere and S. P. Timoshenko, Buckling, in Mechanics of Materials, PWS Publishing Company, Boston (1997) pp. 731±804. 2. R. C. Hibbeler, Mechanics of Materials, third edition, Prentice-Hall, Upper Sadde River (1997). 3. R. R. Craig, Jr., Buckling of Columns, in Mechanics of Materials, John Wiley and Sons, New York (1996) pp. 511±553. 214 J. Neuringer and I. Elishakoff

4. E. P. Popov, Buckling of Columns, in Introduction to Mechanics of Solids, Prentice-Hall, Englewood Cliffs (1968) pp. 515±551. 5. S. H. Crandall, N. C. Dahl and T. J. Lardner, Stability of Equilibrium: Buckling, in An Introduction to the Mechanics of Solids, McGraw-Hill Kogakusha, Tokyo (1978) pp. 577±612. 6. S. P. Timoshenko and J. M. Gere, Theory of Elastic Stability, McGraw-Hill, New York (1963). 7. A. Chajes, Principles of Structural Stability Theory, Prentice-Hall, Engelwood Cliffs (1974). 8. D. O. Brush and B. O. Almroth, Buckling of Bars, Plates and Shells, McGraw-Hill, New York (1975). 9. G. Simitses, Introduction to the Elastic Stability of Structures, Prentice Hall, Englewood Cliffs (1976). 10. Z. Bazant and L. Gedolin, Stability of Structures, Oxford University Press, New York (1991). 11. A. Gajewski and M. Zyczkowski, Optimal Structural Design under Stability Constraints, Kluwer Academic Publishers, Dortrecht (1988). 12. R. T. Haftka and Z. GuÈrdal, Elements of Structural Optimization, Kluver Academic Publishers, Dortrecht (1992). 13. B. Budiansky and J. Hutchinson, Buckling: Progress and Challenge, Trends in Solid Mechanics (J. F. Besseling and A. M. A. van der Heijden, eds.), Delft University Press, (1979) pp. 93±116. 14. N. Olhoff and J. E. Taylor, Designing continuous columns for minimum total cost of material and interior supports, J. Structural Mechanics, 6, (1978) pp. 367±382. 15. G. I. N. Rozvany, and Mroz Z., Column design: optimization of support conditions and segmentation, J. Structural Mechanics, 5, (1977) pp. 279±290. 16. D. Szelag and Z. Mroz, Optimal Design of Vibrating Beams with Unspecified Support Reactions, Computer Methods in Applied Mechanics and Engineers, 19, (1979) pp. 333±349. 17. B. P. Wang, Eigenvalue sensitivity with respect to location of internal stiffness and mass attachments, AIAA Journal, 31, (1993) pp. 791±794. 18. C. M. Wang, K. M. Liew, L. Wang and K. K. Ang, Optimal locations of internal line supports for rectangular plates against buckling, Structural Optimization, 4, (1992) pp. 199±205. 19. C. M. Wang, L. Wang, K. K. Ang and K. M. Liew, Optimization of internal line support positions for plates against vibration, Mechanics of Structures and Machines, 21, (1993) pp. 429±454. 20. Z. S. Liu, H. C. Hu and D. J. Wang, Effect of small variation of support location on natural frequencies, Proc. Int. Conf. Vibration Engineering, ICVE'94, International Academic Publishers, Beijing, (1994) pp. 9±12. 21. Z-S. Liu, H-C. Hu. and D-J. Wang, New method for deriving eigenvalue rate with respect to support location, AIAA Journal, 34, (1996) pp. 864±866. 22. J. W. Hou and C. H. Chuang, Design sensitivity analysis and optimization of vibrating beams with variable support locations, 16th Automation Conference, Chicago, IL, ASME, New York, (1990). 23. J. Ari-Gur and I. Elishakoff, Influence of the shear deformation on the buckling of a structure with overhang, J. Sound and Vibration, 139, (1990) pp. 165±169.

APPENDIX A

Derivation of equation (38) The general solution for w1 x† is given by equation (11). The condition w1 0† ˆ 0 yields A4 ˆ A2. We calculate the slope:

0 w1 x† ˆ A1k cos kx A2 k sin kx ‡ A3 A1† 0 The condition w1 0† ˆ 0 results in:

A3 ˆ A1 k A2† Thus, expression for the displacement becomes:

w1 x† ˆ A1 sin kx kx† ‡ A2 cos kx 1† A3†

Imposition of the condition w1 a† ˆ 0 yields:

A1 sin ka ka† ‡ A2 cos ka 1† ˆ 0 A4† or sin ka ka A ˆ A A5† 2 1 cos ka 1

Substituting for A2 results in:

w1 x† ˆ A‰ cos ka 1† kx sin kx† ‡ cos kx 1† sin ka ka†Š A6† with A being a new constant:

A ˆ A1= cos ka 1† A7† Interesting Instructional Problems in Column Buckling 215

Multiplying out in equation (A6) and using the identity: sin kx† cos ka† cos kx† sin ka† ˆ sin k x a† A8† we obtain:

w1 x† ˆ A‰ sin k x a† sin ka ka cos kx ‡ kx cos ka ‡ sin kx k x a†Š A9†

APPENDIX B

Derivation of equation (39) The general solution for w2 x† is given by equation (13). The imposition of the condition w2 L† ˆ 0 yields:

B1 sin kL ‡ B2 cos kL ‡ B3L ‡ B4 ˆ 0 B1† Now, the expression for the slope reads:

0 w2 x† ˆ B1 k cos kx B2 k sin kx ‡ B3 B2† 0 The condition w2 L† ˆ 0 results in:

B1 k cos kL B2 k sin kL ‡ B3 ˆ 0 B3†

The condition w2 a† ˆ 0 yields:

B1 sin ka ‡ B2 cos ka ‡ B3 a ‡ B4 ˆ 0 B4† We have from equations (B2), (B3) and (B4):

B2 cos kL ‡ B3 L ‡ B4 ˆ B1 sin kL

B2 k sin kL ‡ B3 ‡ B4 ? 0 ˆ B1 k cos kL B5†

B2 cos ka ‡ B3 a ‡ B4 ˆ B1 sin ka Solving these equations using Cramer's rule, we get:

B2 ˆ B1 '2='1

B3 ˆ B1 '3='1 B6†

B4 ˆ B1 '4='1 where

cos kL L 1

'1 ˆ k sin kL 1 0

cos ka a 1

sin kL L 1

'2 ˆ k cos kL 1 0

sin ka a 1 B7†

cos kL sin kL 1

'3 ˆ k sin kL k cos kL 0

cos ka sin ka 1

cos kL L sin kL

'4 ˆ k sin kL 1 k cos kL

cos ka a sin ka 216 J. Neuringer and I. Elishakoff

Substituting for B2, B3 and B4 in terms of B1 (equation B6), simplifying and introducing a new constant we obtain equation (39). As a check, by directly substituting the appropriate values for x in equation (39) it is 0 immediately verified that equation (39) satisfies the conditions w2 L† ˆ w2 L† ˆ w2 a† ˆ 0.

Joseph L. Neuringer is a Mathematical Phycisist. He obtained his PhD degree in Physics from New York University in 1951. He was Chief Scientist of the Plasma Propulsion Laboratory at Republic Aviation Corporation until 1962 thereafter joining the Avco Systems Division as Senior Consulting Scientist until 1970. From 1970 to retirement in 1985 he was Professor of Mathematics at the University of Massachusetts (Lowell). His research interests and publications are in Plasma Physics, Reentry Physics, Magneto- hydrodynamics, Ferrohydrodynamics, High Speed Gas Dynamics, High Temperature Gas Effects on Structures, and the Scattering of Electromagnetic Radiation by Small Particles and by Random Turbulent Media. Although retired, Dr. Neuringer found a new passion in attending numerous classes ranging from the engineering sciences to humanities, at the Florida Atlantic University.

Isaac Elishakoff serves as a Professor in the Department of Mechanical Engineering, and Mathematics. He got his PhD from the Moscow Power Engineering Institute and State University in Moscow, Russia in 1971. His interest in the innovative teaching methods dates to 1970, when he graduated from the University of Pedagogical Performance at the Moscow Power Engineering Institute & State University, in parallel to pursuing his PhD degree. He is a Fellow of the American Academy of Mechanics, and of the Japan Society of Promotion of Sciences. He serves as a General Advisory Editor to Elsevier Science Publishers in Amsterdam, the Netherlands, and is an Associate Editor of three Journals. He also serves on editorial boards of five other international journals. He was a faculty member in the Technion-Israel Institute of Technology during 1972±1989 (Professor of Aeronautical Engineering since 1984). He also held visiting positions as a Freimann Chair Professor, and then as Massman Chair Professor at the University of Notre Dame; Distinguished Alberto Castigliano Professor in the University of Palermo, Italy; Professor at the Delft University of Technology, in the Netherlands as well as in the Naval Postgraduate School in California. Dr Elishakoff is an author or co-author of four books and over 220 research papers; and co-editor of ten books. Presently he serves as a ASME Distinguished Lecturer.