3 Engineering Example 5

3.1 Buckling of a strut

The equation governing the buckling load P of a strut with one end fixed and the other end simply supported is given by tan μ L = μ L where μ = P E I , L is the length of the strut and E I is the flexural rigidity of the strut. For safe design it is important that the load applied to the strut is less than the lowest buckling load. This equation has no exact solution and we must therefore use the method described in this Workbook to find the lowest buckling load P .

Figure 26

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We let μ L = x and so we need to solve the equation tan x = x . Before starting to apply the Newton-Raphson iteration we must first obtain an approximate solution by plotting graphs of y = tan x and y = x using the same axes.

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From the graph it can be seen that the solution is near to but below x = 3 π 2 ( 4.7 ) . We therefore start the Newton-Raphson iteration with a value x 0 = 4.5 .

The equation is rewritten as tan x x = 0 . Let f ( x ) = tan x x then f ( x ) = sec 2 x 1 = tan 2 x

The Newton-Raphson iteration is x n + 1 = x n tan x n x n tan 2 x n , x 0 = 4.5

so x 1 = 4.5 tan ( 4.5 ) 4.5 tan 2 4.5 = 4.5 0.137332 21.504847 = 4.493614   to 7 sig.fig.

Rounding to 4 sig.fig. and iterating:

x 2 = 4.494 tan ( 4.494 ) 4.494 tan 2 4.494 = 4.494 0.004132 20.229717 = 4.493410   to 7 sig.fig.

So we conclude that the value of x is 4.493 to 4 sig.fig. As x = μ L = P E I L we find, after re-arrangement, that the smallest buckling load is given by P = 20.19 E I L 2 .

Exercises
  1. By sketching the function f ( x ) = x 1 sin x show that there is a simple root near x = 2 . Use two iterations of the Newton-Raphson method to obtain a better estimate of the root.
  2. Obtain an estimation accurate to 2 d.p. of the point of intersection of the curves y = x 1 and y = cos x .
  1. x 0 = 2 , x 1 = 1.936 , x 2 = 1.935
  2. The curves intersect when x 1 cos x = 0 . Solve this using the Newton-Raphson method with initial estimate (say) x 0 = 1.2 .

    The point of intersection is ( 1.28342 , 0.283437 ) to 6 significant figures.