Question 16.CS.1: In Sec. 16.1A, we derived the critical load for an elastic p...

MODELING: As previously discussed, the out-of-straightness of a pin-ended column will be modeled as depicted by the free-body diagram shown in CS Fig. 16.1. This is identical to the column modeled by Fig. 16.7, except that now the total deflection at any location x will now be $(y + y_0)$. Considering the equilibrium of a free-body diagram of the upper portion of this column, similar to that shown in Fig. 16.7b, the bending moment at any location x will now be $M = −P (y + y_0)$.

ANALYSIS: For the column as modeled, substituting the bending moment obtained into Eq. (16.4) gives

$\frac{d^2 y}{d x^2}=\frac{M}{E I}=-\frac{P}{E I} y$      (16.4)

Rearranging, we get

$\frac{d^2 y}{d x^2}+\frac{P}{E I} y=-\frac{P}{E I} y_0$     (1)

As stated earlier, while the actual out-of-straightness will generally not follow a simple curve, upon loading the overall deflected shape will eventually approach that for an eccentrically loaded, initially straight column. Assuming, then, that the initial out-of-straightness is also symmetric, and noting that $y_0$ is zero at the ends, we will define $y_0$ as

$y_0=\delta_0 \sin \frac{\pi x}{L}$     (2)

where $\delta_0$ is the initial deformation at the center of the column. Setting

Equation (1) can now be written as

$\frac{d^2 y}{d x^2}+p^2 y=-p^2 \delta_0 \sin \frac{\pi x}{L}$     (3)

The solution for Eq. (3) is then

$y=A \sin p x+B \cos p x+\frac{1}{\frac{\pi^2}{p^2 L^2}-1} \delta_0 \sin \frac{\pi x}{L}$     (4)

Because y = 0 at x = 0, B = 0. Then, substituting y = 0 at x = L gives A = 0. The equation for the deflection is thus

$y=\frac{1}{\frac{\pi^2}{p^2 L^2}-1} \delta_0 \sin \frac{\pi x}{L}$     (5)

From Eq. 16.11a, $P_{cr}=\pi^2 E I / L^2$. We let

$P_{cr}=\frac{\pi^2 E I}{L^2}$     (16.11a)

We can then write the deflection as

$y=\frac{1}{\frac{1}{\beta}-1} \delta_0 \sin \frac{\pi x}{L}=\frac{\beta}{1-\beta} \delta_0 \sin \frac{\pi x}{L}$     (6)

Equation (6) gives the additional deflection due to the load. Adding this to the initial deflection (or out-of-straightness), the total deflection is

$y+y_0=\frac{\beta}{1-\beta} \delta_0 \sin \frac{\pi x}{L}+\delta_0 \sin \frac{\pi x}{L}=\frac{1}{1-\beta} \delta_0 \sin \frac{\pi x}{L}$     (7)

At the center of the column, the total deflection $(\delta)_{x=L / 2}$ is then

$(\delta)_{x=\frac{L}{2}}=\delta_0 \frac{1}{1-\beta}=\delta_0 \frac{1}{1-\frac{P}{P_{cr}}}$     (8)

Equation (8) shows that the deflection at the center is larger than $\delta_0$, which was the initial deformation at the column center prior to loading. The plot of this equation will be similar to the plot in CS Fig. 16.2. In applying this result to the testing of an actual column, let us now consider the measured deflection at the center of the column as a function of the load taken in incremental steps during the testing. This displacement is equivalent to the total displacement given in Eq. (8) minus the initial displacement defined by Eq. (2) with x = L/2. This displacement $\delta_{\text {test }}$ is then

$\left(\delta_{\text {test }}\right)_{x=\frac{L}{2}}=\delta_0\left(\frac{1}{1-\frac{P}{P_{cr}}}-1\right)=\delta_0\left(\frac{\frac{P}{P_{cr}}}{1-\frac{P}{P_{cr}}}\right)=\delta_0\left(\frac{1}{\frac{P_{cr}}{P}-1}\right)$     (9)

If we eliminate the denominator in Eq. (9), we get

$\left(\delta_{\text {test }}\right)_{x=\frac{L}{2}}\left(\frac{P_{cr}}{P}-1\right)=\delta_0$     (10)

This equation can be rewritten as

$\frac{\left(\delta_{\text {test }}\right)_{x=\frac{L}{2}}}{P}=\frac{1}{P_{cr}}\left(\delta_0+\left(\delta_{\text {test }}\right)_{x=\frac{L}{2}}\right)$     (11)

This is the equation of a straight line. We plot $\frac{\left(\delta_{\text {test }}\right)_{x=\frac{L}{2}}}{P}$ on the vertical axis and $\left(\delta_{\text {test }}\right)_{x=\frac{L}{2}}$ on the horizontal axis. The line crosses the x axis at $-\delta_0$ and the slope is $\frac{1}{P_{cr}}$. Thus, the critical load is the inverse of the slope of the plotted line.

The method we have just developed for determining the critical load from experimental data obtained for real columns, i.e., those with initial out-of-straightness, was first proposed by Southwell.${}^i$ The approach produces excellent results for columns that are in the elastic range, and in which the initial out-of-straightness is small. Later researchers have found that it can also be used for columns that are not loaded perfectly through the column center, i.e., columns with small eccentric loads.

REFLECT and THINK: Southwell used the data from tests conducted by T. von Kármán to assess the accuracy of his approach. The tests were carefully carried out to ensure precise centering of the loads on the ends of the pin-ended columns made of mild steel with short, medium, and long lengths. A total of eight columns were tested.

The columns had rectangular cross sections, and the slenderness ratios L/r ranged from approximately 91 to 176.

The plots for two of the sets of data are shown in CS Fig. 16.3, one for a shorter column (L/r = 95) and one for a longer column (L/r = 116).

All eight plots are similar. As can be seen for each test example, a straight line can be drawn approximately through the data points. The line through the data points will not go through the intersection of the two axes, but instead should cross the horizontal axis at a point equivalent to the initial deflection prior to loading at the center of the column.

Southwell determined the critical loads using curve-fitting algorithms for the plots of all eight columns. The resulting ratio of the critical load estimated from the test using the plots versus the critical load determined with Euler’s formula given by Eq. (16.11a) ranged from 0.980 to 1.022 for the eight columns. Thus, the results from the test estimations were within approximately 2% of those determined from Euler’s formula.

Real columns have imperfections, including both initial out-of-straightness and loads that are not perfectly centered on the end cross section. Southwell’s approach can be used to account for both types of imperfections, provided that the column is fully elastic and provided that the initial out-of-straightness is not large. This approach has been widely applied in research investigations seeking to determine critical loads from real tests. It was undoubtedly used by the researchers who tested the column in Photo 16.1.

${}^i$ See R. V. Southwell, “On the Analysis of Experimental Observations in Problems of Elastic Stability,” Proceedings of the Royal Society, 1932, Volume 135, pp. 601‒616.