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Columns and Other Compression Members Chapter 12 Columns and Other Compression Members 12.1 Columns; Common Case The formulas and discussion of this section are based on the following assumptions: (1) The column is nominally straight and is subjected only to nominally concentric and axial end loads, and such crooked- ness and eccentricity as may occur are accidental and not greater than is consistent with standard methods of fabrication and ordinary conditions of service; (2) the column is homogeneous and of uniform cross section; (3) if the column is made up of several longitudinal elements, these elements are so connected as to act integrally; (4) there are no parts so thin as to fail by local buckling before the column as a whole has developed its full strength. End conditions. The strength of a column is in part dependent on the end conditions, that is, the degree of end fixity or constraint. A column with ends that are supported and fixed, so that there can be neither lateral displacement nor change in slope at either end, is called fixed- ended. A column with ends that are supported against lateral dis- placement but not constrained against change in slope is called round-ended. A column with one end fixed and the other end neither laterally supported nor otherwise constrained is called free-ended. A column with both end surfaces that are flat and normal to the axis and that bear evenly against rigid loading surfaces is called flat-ended. A column with ends that bear against transverse pins is called pin-ended. Truly fixed-ended and truly round-ended columns practically never occur in practice; the actual conditions are almost always interme- diate. The greatest degree of fixity is found in columns with ends that are riveted or welded to relatively rigid parts that are also fixed. 525 526 Formulas for Stress and Strain [CHAP.12 Theoretically a flat-ended column is equivalent to a fixed-ended column until the load reaches a certain critical value at which the column ‘‘kicks out’’ and bears only on one edge of each end surface instead of on the whole surface. Actually, flat-ended columns have a degree of end constraint considerably less than that required to produce fixity. The nearest approach to round-ended conditions is found in pin-ended columns subject to vibration or other imposed motion. The degree of end fixity may be expressed by the coefficient of constraint [explained following Eq. (12.1-1)] or by the free or effective length, which is the length measured between points of counterflexure or the length of a round-ended column of equal strength. Behavior. If sufficiently slender, a column will fail by elastic instabil- ity (see Chap. 15). In this case the maximum unit stress sustained is less than the proportional limit of the material; it depends on the modulus of elasticity, the slenderness ratio (ratio of the length of the column to the least radius of gyration of the section), and the end conditions and is independent of the strength of the material. Columns which fail in this way are called long columns. Columns that are too short to fail by elastic instability are called short columns; such a column will fail when the maximum fiber stress due to direct compression and to the bending that results from accidental crookedness and eccentricity reaches a certain value. For structural steel this value is about equal to the tensile yield point; for light alloys it is about equal to the compressive yield strength; and for wood it lies between the flexural elastic limit and the modulus of rupture. For a given material and given end conditions, there is a certain slenderness ratio which marks the dividing point between long and short columns called the critical slenderness ratio. Formulas for long columns. The unit stress at which a long column fails by elastic instability is given by the Euler formula P Cp2E ¼ ð12:1-1Þ A ðL=rÞ2 where P ¼ total load, A ¼ area of section, E ¼ modulus of elasticity, L=r ¼ slenderness ratio, and C is the coefficient of constraint, which depends on end conditions. For round ends, C ¼ 1; for fixed ends, C ¼ 4; and for the end conditions that occur in practice, C can rarely be assumed greater than 2. It is generally not considered good practice to employ long columns in building and bridge construction, but they are used in aircraft. (Formulas for the loads producing elastic instabil- ity of uniform and tapered bars under a wide variety of conditions of loading and support are given in Table 15.1.) SEC. 12.1] Columns and Other Compression Members 527 Formulas for short columns. It is not possible to calculate with accu- racy the maximum stress produced in a short column by a nominally concentric load because of the large influence of the indeterminate crookedness and eccentricity. The maximum unit stress that a column will sustain, however, can be expressed by any of a number of formulas, each of which contains one or more terms that is empirically adjusted to secure conformity with test results. Of such formulas, those given below are the best known and provide the basis for most of the design formulas used in American practice. In these equations P denotes the load at failure, A the cross-sectional area, L the length, and r the least radius of gyration of the section; the meaning of other symbols used is explained in the discussion of each formula. Secant formula P s ¼ rffiffiffiffiffiffiffi! ð12:1-2Þ A ec KL P 1 þ sec r2 2r AE This formula is adapted from the formula for stress due to eccentric loading [Eq. (12.4-1)]. Here s denotes the maximum fiber stress at failure (usually taken as the yield point for steel and as the yield strength for light alloys); e denotes the equivalent eccentricity (that eccentricity which in a perfectly straight column would cause the same amount of bending as the actual eccentricity and crookedness); c denotes the distance from the central axis about which bending occurs to the extreme fiber on the concave or compression side of the bent column; and K is a numerical coefficient, dependent on end conditions, such that KL is the effective length of the column, or distance between points of inflection. The term ec=r2 is called the eccentric ratio, and a value is assumed for this ratio which makes the formula agree with the results of tests on columns of the type under consideration. For example, tests on structural steel columns of conventional design indicate that the average value of the eccentric ratio is 0.25. In using the secant formula, P=A must be solved for by trial or by the use of prepared charts. Rankine formula P s ¼ ð12:1-3Þ A 1 þ fðL=rÞ2 This is a semirational formula. The value of s is sometimes taken as the ultimate strength of the material and the value of f as s=Cp2E, thus making the formula agree with the results of tests on short prisms when L=r is very small and with Euler’s equation when L=r is 528 Formulas for Stress and Strain [CHAP.12 very large. More often s and f are adjusted empirically to make the equation agree with the results of tests through the L=r range of most importance. Simple polynomial formula P L n ¼ s À k ð12:1-4Þ A r This is an empirical formula. For most steels the exponent n is chosen as 2 and s is chosen as the yield point to give the well-known parabolic formula. The constant k is generally chosen to make the parabola intersect tangent to the Euler curve for a long column. See Refs. 1 to 4. For cast irons and for many of the aluminum alloys (Refs. 1 and 5) the exponent n is taken as unity to give a straight-line formula. If s were here taken as the maximum fiber stress at failure and the straight line made tangent to the Euler curve, the formula would give values well below experimental values. For this reason the straight-line formula is generally used for intermediate lengths with s and k modified to make the straight line pass through the experi- mental data of interest. For columns shorter than intermediate in length, a constant value of P=A is given; for those longer than intermediate, the Euler curve is specified. For timber the exponent n is usually chosen as 4 (Refs. 1 and 6) and then treated in the same manner as was the parabolic formula. The value used for s is generally the ultimate compressive strength of the timber. Exponential formula P 2 KL s 1=2 ¼ Cl s where l ¼ ð12:1-5Þ A 1 rp E The American Institute of Steel Construction in Ref. 7 suggests the use of a formula of the form shown in Eq. (12.1-5), where K, L, r, and s are as defined for the secant formula. The constant l combines the column dimensions L and r, the degree of end fixity indicated by K, and the material properties s and E. The secant, Rankine, parabolic, exponential, and Euler formulas can all be expressed in terms of l and a simple tabulation used to compare them. P=A l2 Parabolic formula ¼ 1 À for l < 1:414 s 4 P=A 2 Exponential formula ¼ 0:6922l for l < 1:649 s SEC. 12.2] Columns and Other Compression Members 529 P=A 1 Rankine formula ¼ for all l s 1 þ l2 P=A 1 Secant formula ¼ rffiffiffiffiffiffiffiffiffiffi!for all l s pl P=A 1 þ 0:25 sec 2 s The Euler formula becomes ðP=AÞ=s ¼ 1=l2, and both the parabolic and exponential formulas given above have been derived to be tangent to the Euler formulas where they intersect.
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