Module 10

Columns

Objectives:

When you finish this module (especially the appropriate textbook sections and problems), you will be able to:

· determine the stability of a column under a specified load

· determine the critical load for a column

· use the secant formula for eccentric loading

· design a column for centric or eccentric loading

Assignments: 10.4, 11, 13, 17, 21, 32, 42, 51, 58, 66, 89, 90, 95, 101

Reference: §10.1 - 10.7 of Mechanics of Materials, 3rd Edition, Beer, Johnston, & DeWolf, McGraw-Hill

Euler's Formula: Your textbook first considers a simplified model of a column - two rigid rods pin-connected at point C (see Figure 10.4, page 609), under a load P, constrained at A and B to move vertically, and with the ability to flex (torsional spring of constant K at C). After doing an analysis of the free bodies (Figure 10.5, 6, & 7), the text develops equation 10.3, which is then used to develop Euler's formula (equation 10.11, page 611):

For a load P < P_cr (the critical load), the column will not buckle. As usual, E is modules of elasticity and I is the cross-sectional moment of inertia. L is the length of the column. This formula is often written in terms of the critical stress:

L/r is called the slenderness ratio; r is the radius of gyration for the cross section. In using these equations, the minimum values of I and r should be used. Figure 10.9 (page 613) plots the stress vs the slenderness ratio. As the ratio decreases (for example, r might be held constant while the column length is decreased), the possible critical stress that will cause buckling increases. Of course, one cannot exceed the yield stress.

The above were developed for columns which are pin-connected at both ends and are constrained to move vertically (see Fig. 10.2, page 608). This is the same as case b in Figure 10.18 (page 617), where L_e (effective length) is the same as the actual length. The other cases shown in that figure are marked so that you can identify where the equivalent pin connections would be. For example, in case a the fixed end (B) cannot move and is effectively the midpoint in a column whose effective length is twice the actual length. The other end is free (not constrainedvertically). In contrast, case c's top end is constrained vertically and so the effective length is 0.7L.

Sample Problem 10.1 (page 618) shows how Euler's formula is used for an actual column:

· The radius of gyration is computed from the moment of inertia and the area

· The slenderness ratio is then calculated in terms of L, noting that the situation fits case c in Figure 10.18 (eqn. 1)

· Because the cross section is rectangular, the moment of inertia I_y is different from I_z, and r_y is different from r_z.

Furthermore, the top end is free in the z-direction, and we use case a in Figure 10.18, getting equation 2. You see, we have to consider the possibility of buckling in two directions!

· If the column is designed considering only one direction of buckling, it may a) either buckle in the other direction, or

b) be overdesigned in the other direction. By equating (1) and (2), we demand that the critical stresses (see equation 10.13 on page 612) are equal in both directions, thus obtaining the most efficient design.

· Finally, the authors use the given data, including the factor of safety, to find the dimensions of the rectangular column.

Eccentric Loading: The load P is not always applied centrally (see Figure 10.19a, page 625, where e is the eccentricity; i.e., the distance from the center that the load is applied). This has the effect of applying an additional bending moment to the column; y_max is the maximum sideways displacement which in this case is shown at the center of the beam. The problem is solved like the beam problems were - writing the differential equation (page 626) and solving it:

After a page or two of work and discussion, and checking to see that we get Euler's formula for a centric load, your text obtains equation 10.31:

P_cr is computed using Euler's formula. Equation 10.35 provides a means for calculating the maximum normal stress. Your authors point out that the stress does not vary linearly with the load P. Therefore, if there are several loads, you cannot add the displacements caused by them to get y_max. Instead, you must first compute the resultant load and then use equation 10.34 or 10.35 to find the stress. Likewise, the factor of safety must be applied to the load, not the stress!

The secant formula is equation 10.36 on page 628. It was obtained by rewriting equation 10.34 in terms of the slenderness ratio. But P/A appears on both sides, and trial and error methods are often necessary to solve this when specific data are given. To simplify the work, Figure 10.24 has been provided. But here's an example involving temperature expansion for which trial and error methods are not needed!