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from the right end. We evaluate F(x) at the chosen points. If the value of F(x) at x(1) is algebraically smaller than at x(2), then we eliminate the subinterval between x(2) and the right end of the search interval. In the oppositive event, we eliminate the subinterval between x(1) and the left end. The distance of 0.236D between x(1) and x(2) equals 0.382 of the new search interval, which is 0.618D overall. Thus we have already located one point at 0.382 of the new search interval, and we have determined the value of F(x) at that point. We need only locate a point that is 0.382 from the opposite end of the new search interval, evaluate F(x) at that point, and go through the elimination procedure, as before, to further reduce the search interval. The process is repeated until the entire remaining interval is small and the value of F(x) at the two points at which it is evaluated is the same within the desired accuracy. The preceding method may run into difficulty if the function F(x) is flat, that is, if F(x) does not vary over a substantial part of the search interval or if F(x) happens to have the same value at the two points that are compared with each other. The first case is not likely to occur in the types of problems under consideration here. In the second case, we calculate F(x) at another point, near one of the two points of comparison, and use the newly chosen point in place of the nearby original point. Reference [30.2] derives the golden-section search strategy and provides equations (p. 289) for predicting the number of function evaluations required to attain a specified fractional reduction of search interval or an absolute final search interval size.

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NCD-106-3

The reason why we devoted so much attention to uniform prismatic column problems is that their solution is analytically simple, so that we could obtain the results directly. In most other cases, we have to be satisfied with approximate formulations for computer (or programmable calculator) calculation of the solution. The finiteelement method is the approximation method most widely used in engineering at

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the present time to reduce problems dealing with continuous systems, such as beams and columns, to sets of algebraic equations that can be solved on a digital computer. When there is a single independent variable involved (as in our case), the interval of interest of the independent variable is divided into a set of subintervals called finite elements. Within each subinterval, the solution is represented by an arc that is defined by a simple function, usually a polynomial of low degree. The curve resulting from the connected arc segments should have a certain degree of smoothness (for the problems under discussion, the deflection curve and its first derivative should be continuous) and should approximate the solution. An effective method of obtaining a good approximation to the solution is based on the mechanical energy involved in the deformation process.

NCD-106-2

Beams that have rectangular cross sections with the thickness much smaller than the depth are prone to instability involving rotation of the beam cross section about the beam axis.This tendency to instability arises because such beams have low resistance to torsion about their axes. In preparation for the analysis of this problem, recall that for a circular cylindrical member of length L and radius R, subjected to an axial torque T, the angle of twist is given by = TL JG (30.25)

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where G = shear modulus and J = polar moment of inertia of the cross section. For noncircular cross sections, the form of the right side of Eq. (30.25) does not change; the only change is in the expression for J, the torsion constant of the cross section. If the thickness of the cross section, to be denoted by t, is small and does not vary much, then J is given by

3J =

(30.26)

3 4 5 6 7 8 9 16 15 14 13 12 11 10

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