Columns lateral buckling of baks compressed within the elastic limit

There is a certain critical value of the compressive force at which large lateral deflection may be produced by the slightest later load. For a prismatical bar built-in at the bottom and loaded axially at the top (Fig. 21a) this critical compressive force is

P_cr=² EI/4l^2 (27)

From the equation above this critical load will be seen not to de­pend on the strength of the material but only upon the dimensions of the structure and the modulus of elasticity of the material. Two equal slender struts, one of high strength steel and the other of common structural steel, will buckle at the same compressive force, although the strength of the material in the two cases is very differ­ent. Equation (27) shows also that the strength of a strut may be raised by increasing I. This may be done without changing the cross

sectional area by distributing the material as far as possible from the principal axes of the cross section. Hence, tubular sections are more economical than solid for compression members.

Diminishing the wall thickness of such sections and increasing the transverse dimensions can increase their stability. There is a lower limit for the wall thickness, however, below which the wall itself becomes unstable and instead of buckling of the strut аs а whole, there is a buckling of its longitudinal elements, which brings about a corrugation of the wall.

This discussion shows then the sidewise buckling of compression members, i. e., and their elastic stability to be of great practical impor­tance. This is especially true in many modern structures where the cross section dimensions are being made smaller and smaller due to the use of stronger materials and the desire to save weight. In many-cases failure of an engineering structure is to be attributed to elas­tic instability and not to the lack of strength on the part of the ma­terial. Let us consider the case mentioned above (Fig. 21a).

If the load P is less than its critical value the bar remains straight and undergoes only axial compression. This straight form of elastic equilibrium is stable, i. e., if a lateral force were applied and a small deflection produced, this deflection disappears when the lateral force is removed and the bar becomes straight again. By increasing P grad­ually we may arrive at a condition in which the straight form of equilibrium becomes unstable and a slight lateral force may produce a lateral deflection, which does not disappear with the cause, which produces it. The critical load is then defined as the axial load, which is sufficient to keep the bar in a slightly bent form (Fig. 21a).

Euler's Column Formulas for Other End Conditions

The critical load for some other cases can easily be obtained from the solution for the foregoing case. For example, in the case of a bar with hinged ends (sometimes called pivot, round, or pin ends) shown in Fig. 21b, it is evident from symmetry that each half of the bar is in the same condition as the entire bar of Fig. 21a. Hence the criti­cal load for this case is obtained by using 1/2, instead of 1, in eq. (27)

P_cr=² EI/l^2 (28)

The case of a bar with hinged ends is very often encountered in prac­tical applications and is called the fundamental case of buckling of a prismatic bar. In the case of a bar with built-in (or fixed) ends, shown in Fig. 21c, there are' reactive moments which keep the ends from rotating during buckling. The combination of the compressive force with these moments is equivalent to the compressive force P, Fig. 21c, applied eccentrically. There are inflection points where the line of action of P intersects the deflection curve because the bending mo­ments at these points are zero. These points and the midpoint of the span divide the bar into four equal portions, each of which is in the same condition as the bar represented in Fig. 21a. Hence the critical load for a bar with built-in ends is found from eq. (27), by using 1/4 instead of 1, which gives

P_cr=4² EI / l^2 (29)

It was assumed in the previous discussion that the bar is very slender so that the stress accompanying the bending, which occurs during buckling, remains within the proportional limit. To establish the limit of applica­bility of the above formulas for critical loads, consider the fundamen­tal case (Fig. 21b). Divide eq. (28) by the cross-sectional area A of the bar, and let r = I/A be the smaller radius of gyration. Then the critical value of the compressive stress is

s_cr= P_cr / A (30)

This equation is applicable as long as the stress s_cr remains within the proportional limit. With this limit and also the modulus E known for a given material, the limiting value of the ratio I/r (which charac­terizes the slenderness of the bar) can easily be obtained from eq. (30) for each particular case. For structural steel with a proportional limit 30,000 lbs. per sq. in., and E = 30Xl0⁶ lbs. per sq. in., we find from eq. (30)

L / r100 (31)

Consequently the critical load for steel bars hinged at the ends, having l/r=100, is²³ to be calculated from eq. (30). When l/r=100 the compressive stress reaches the limit of proportionality before buckling can occur, and the equation de­rived above on the basis of perfect elasticity is no lon­ger applicable.

Equation (30) may be represented graphically. The curve approaches the axis of abs­cissas asymptotically; that is, the critical stress diminishes indefinitely with increase in the slenderness of the bar. The curve also approaches the vertical axis asymptotically but here it is applicable only so long as the stress s_cr is below the proportional limit of the material. In the case of the above structural steel, for instance, the point С is the upper limit of applicability of the curve.

The equation for the fundamental case (eq. 30) may be used for these cases also if we use a modified length l instead of the length of the bar. In the case of a prismatic bar with one end built in and the other free, the modified length is twice as great as the actual length l₁ = 2l. In the case of a prismatic bar with both ends built in, the modified length is half the actual length, l₁ = ¹/₂l. The equa­tion for the critical stress may consequently be represented in the form:

s_cr=²E(r/ l₁)² (33)

in which  depends upon the conditions at the ends of the bar and is sometimes called the length coefficient. In the discussion, which follows, only the fundamental case of buckling is considered. The results obtained may be used for other cases of buckling of bars by taking the modified length instead of the actual length.