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Note that the sinusoidal shape of the solution function is determined by the differential equation and does not depend on the boundary conditions. If we can find a segment of a sinusoidal curve that satisfies our chosen boundary conditions and, in turn, we can find some segment of that curve which matches the curve in Fig. 30.1, we can establish a correlation between the two cases. This notion is the basis for the effective-length concept. Recall that Eq. (30.9) was obtained for a column with both ends simply supported (that is, the moment is zero at the ends). Figure 30.2 illustrates columns of length L with various idealized end conditions. In each case, there is a multiple of L, KL, which is called the effective length of the column Leff, that has a shape which is similar to and behaves like a simply supported column of that length. To determine the critical loads for columns whose end supports may be idealized as shown in Fig. 30.2, we can make use of Eq. (30.9) if we replace L by KL, with the appropriate value of K taken from Fig. 30.2. Particular care has to be taken

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IBOutlet UITextView *messageTextView; IBOutlet UILabel*deviceTokenField; IBOutlet UITextField *usernameField; -(IBAction)handleSendButton:(id)sender;

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.

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to distinguish between the case in Fig. 30.2c, where both ends of the column are secured against rotation and transverse translation, and the case in Fig. 30.2e, where the ends do not rotate, but relative transverse movement of one end of the column with respect to the other end is possible. The effective length in the first case is half that in the second case, so that the critical load in the first case is four times that in the second case. A major difficulty with using the results in Fig. 30.2 is that in real problems a column end is seldom perfectly fixed or perfectly free (even approximately) with regard to translation or rotation. In addition, we must remember that the critical load is inversely proportional to the square of the effective length. Thus a change of 10 percent in Leff will result in a change of about 20 percent in the critical load, so that a fair approximation of the effective length produces an unsatisfactory approximation of the critical load. We will now develop more general results that will allow us to take into account the elasticity of the structure surrounding the column.

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FIGURE 30.2 Effective column lengths for different types of support. (a) Simply supported, K = 1; (b) fixed-free, K = 2; (c) fixed-fixed, K = 1 2; (d) fixed-pinned, K = 0.707; (e) ends nonrotating, but have transverse translation.

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Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.

Next, switch to Interface builder and add the UITextView, a UILabel, a UITextField (for the username), and a UIButton. Then wire these all up in Interface Builder, as shown in Figure 10-20.

We will begin with a generalization of the case in Fig. 30.2e. In Fig. 30.3, the lower end is no longer free to translate, but instead is elastically constrained. The differential equation is EI d2y = M = M(0) + P[y(0) y] + Hx dx2 (30.10)

Here H, the horizontal force at the origin, may be expressed in terms of the deflection at the origin y(0) and the constant of the constraining spring K(0): H = K(0)y(0) (30.11)

M(0) is the moment which prevents rotation of the beam at the origin. The moment which prevents rotation of the beam at the end x = L is M(L). The boundary conditions are y(L) = 0 dy(L) =0 dx dy(0) =0 dx (30.12)

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The rest of the symbols are the same as before. We define k as in Eq. (30.3). As in the case of the simply supported column, Eqs. (30.10), (30.11), and (30.12) have solutions in which y(x) need not be zero for all values of x, but again these solutions occur only for certain values of kL. Here these values of kL must satisfy Eq. (30.13): [2(1 cos kL) kL sin kL]L3K(0) + EI(kL)3 sin kL = 0 (30.13)

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