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# Monday December 3, 2007 1. Buckling of a tower See gure 8.16 on p. 501 in the text. Consider the following Cauchy-Euler equation1 which describes...

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Homework assignment 4 * Due date: Monday December 3, 2007 1. Buckling of a tower See ﬁgure 8 . 16 on p. 501 in the text. Consider the following Cauchy-Euler equation 1 which describes the deﬂection y ( x ) from the vertical in terms of the height x measured from the top of the untruncated tower: x 2 y 00 + Pa 2 EI y = 0 , x ( a, a + L ) y ( a ) = 0 y 0 ( a + L ) = 0 The top of the tower corresponds to x = a ( a is the part of the tower that is truncated) and the ground corresponds to x = a + L , so the height of the tower is L . The positive parameters are as follows: P for the vertical load, I for the moment of inertia, E for the modulus of elasticity. Under certain conditions -in particular when the load is high enough- the tower may buckle and the purpose of this problem is to determine when this will happen. By deﬁnition, buckling occurs if the above boundary value problem has a nontrivial solution ( y ( x ) 6 = 0). (a) Show that if P EI/ (4 a 2 ), then there is no nontrivial solution which implies that no buckling occurs. Thus, for small enough loads there is no buckling. (b) Assume henceforth that P > EI/ (4 a 2 ). In that case there may be a nontrivial periodic solution as we will discover soon. First show that the ﬁrst boundary condition implies that a nontrivial solution must be of the form: y ( x ) = c x sin ( β ln( x/a )) , where c is a constant to be determined later and β = r Pa 2 EI - 1 4 is a positive constant depending on the load and various parameters. The second bound- ary condition implies that: c (tan ( αβ ) + 2 β ) = 0 , where α = ln ± a + L a ² . This implies that either c = 0 (in which case no buckling occurs), or that tan ( αβ ) = - 2 β. In terms of β , graph the above tangent function and linear function, and show that they intersect inﬁnitely many times for β > 0. For each β at which these graphs intersect (don’t intersect), buckling occurs (does not occur) * MAP 4305; Instructor: Patrick De Leenheer. 1 Here, it is more convenient to assume that solutions have the form y ( x ) = ` x a ´ r for values r satisfying the indicial equation, instead of y ( x ) = x r . 1
(c) Show that the ﬁrst time the above graphs intersect happens at some critical value of β which we denote as β c > 0, and show that β c belongs to the interval ( β 0 , β 1 ), where β 0 = π 2 α , β 1 = π α . Express the critical load P c , which is the minimal load for which buckling occurs, in terms of β c . 2. Determine all real eigenvalues and eigenfunctions of y 00 + λy = 0 , y (0) = 0 , y ( π ) - ay 0 ( π ) = 0 , for all possible values of the positive parameter a > 0. 3. Consider the linear operator L [ y ] = y (4) , deﬁned on the set of functions having continuous derivatives up to order 4, that satisfy the following boundary conditions: y ( a ) = y 0 ( a ) = y ( b ) = y 0 ( b ) = 0 . Show that L is self-adjoint, ie that for all y 1 and y 2 in the domain of L , we have that: ( y 1 , L [ y 2 ]) = ( L [ y 1 ] , y 2 ) . Use this to prove that eigenfunctions of the problem: L [ y ] + λy = 0 , y ( a ) = y 0 ( a ) = y ( b ) = y 0 ( b ) = 0 corresponding to distinct eigenvalues are orthogonal, ie that ( y 1 , y 2 ) = 0 , whenever y 1 and y 2 are eigenfunctions corresponding to distinct eigenvalues. 4. The following equation describes the displacement of an elastic beam which is clamped on both ends x = 0 and x = 1: y (4) + λy = 0 , x (0 , 1) , y (0) = y 0 (0) = y (1) = y 0 (1) = 0 . It can be shown (no need to do this) that the eigenvalues λ are non-positive. Determine all eigenvalues and eigenfunctions. 2

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