Definition of a Slender Column
An eccentrically loaded, pin-ended column is shown in Fig. 12-1a. The moments at theends of the column are
When the loads P are applied, the column deflects laterally by an amount as shown. Forequilibrium, the internal moment at midheight is (Fig. 12-1b)
The deflection increases the moments for which the column must be designed. In thesymmetrical column shown here, the maximum moment occurs at midheight, where themaximum deflection occurs.
Figure 12-2 shows an interaction diagram for a reinforced concrete column. This dia-gram gives the combinations of axial load and moment required to cause failure of a columncross section or a very short length of column. The dashed radial line OA is a plot of theend moment on the column in Fig. 12-1. Because this load is applied at a constant eccen-tricity, e, the end moment, is a linear function of P, given by Eq. (12-1). The curved,solid line OB is the moment at midheight of the column, given by Eq. (12-2). At anygiven load P, the moment at midheight is the sum of the end moment, Pe, and the momentdue to the deflections, The line OA is referred to as a loadmoment curve for the endmoment, while the line OB is the loadmoment curve for the maximum column moment.
Failure occurs when the loadmoment curve OB for the point of maximum momentintersects the interaction diagram for the cross section. Thus the load and moment at fail-ure are denoted by point B in Fig. 12-2. Because of the increase in maximum moment due
Mc = P1e + d2d,
Me = Pe
562 Chapter 12 Slender Columns
Fig. 12-1Forces in a deflected column.
to deflections, the axial-load capacity is reduced from A to B. This reduction in axial-loadcapacity results from what are referred to as slenderness effects.
A slender column is defined as a column that has a significant reduction in its axial-loadcapacity due to moments resulting from lateral deflections of the column. In the derivation ofthe ACI Code, a significant reduction was arbitrarily taken as anything greater than about5 percent [12-1].
Fig. 12-2Load and moment in a column.
Section 12-1 Introduction 563
Buckling of Axially Loaded Elastic Columns
Figure 12-3 illustrates three states of equilibrium. If the ball in Fig. 12-3a is displacedlaterally and released, it will return to its original position. This is stable equilibrium. Ifthe ball in Fig. 12-3c is displaced laterally and released, it will roll off the hill. This isunstable equilibrium. The transition between stable and unstable equilibrium is neutralequilibrium, illustrated in Fig. 12-3b. Here, the ball will remain in the displaced position.Similar states of equilibrium exist for the axially loaded column in Fig. 12-4a. If the col-umn returns to its original position when it is pushed laterally at midheight and released,it is in stable equilibrium; and so on.
Figure 12-4b shows a portion of a column that is in a state of neutral equilibrium.The differential equation for this column is
In 1744, Leonhard Euler derived Eq. (12-3) and its solution,
Fig. 12-3States of equilibrium.
Fig. 12-4Buckling of a pin-ended column.
564 Chapter 12 Slender Columns
Cases with and 3 are illustrated in Fig. 12-4c. The lowest value of will occurwith This gives what is referred to as the Euler buckling load:
Such a column is shown in Fig. 12-5a. If this column were unable to move sidewaysat midheight, as shown in Fig. 12-5b, it would buckle with and the buckling loadwould be
which is four times the critical load of the same column without the midheight brace.Another way of looking at this involves the concept of the effective length of the col-
umn. The effective length is the length of a pin-ended column having the same bucklingload. Thus the column in Fig. 12-5c has the same buckling load as that in Fig. 12-5b. Theeffective length of the column is in this case, where is the length of each of thehalf-sine waves in the deflected shape of the column in Fig. 12-5b. The effective length,
is equal to The effective length factor is Equation (12-4) is generallywritten as
Four idealized cases are shown in Fig. 12-6, together with the corresponding values of theeffective length, Frames a and b are prevented against deflecting laterally. They aresaid to be braced against sidesway. Frames c and d are free to sway laterally when theybuckle. They are called unbraced or sway frames. The critical loads of the columns shownin Fig. 12-6 are in the ratio 1 :4 :1 : 14.
k = 1>n./>n.k/, />2/>2
n = 2,
n = 1.0.Pcn = 1, 2,
n = number of half-sine waves in the deformed shape of the column/ = length of the column
EI = flexural rigidity of column cross section
Fig. 12-5Effective lengths of columns.
Section 12-1 Introduction 565
Fig. 12-6Effective lengths of idealizedcolumns.
Thus it is seen that the restraints against end rotation and lateral translation have amajor effect on the buckling load of axially loaded elastic columns. In actual structures,fully fixed ends, such as those shown in Fig. 12-6b to d, rarely, if ever, occur. This is dis-cussed later in the chapter.
In the balance of this chapter we consider, in order, the behavior and design of pin-ended columns, as in Fig. 12-6a; restrained columns in frames that are braced against lateraldisplacement (braced or nonsway frames), Fig. 12-6b; and restrained columns in frames freeto translate sideways (unbraced frames or sway frames), Fig. 12-6c and d.
Slender Columns in Structures
Pin-ended columns are rare in cast-in-place concrete construction, but do occur in precastconstruction. Occasionally, these will be slender, as, for example, the columns supportingthe back of a precast grandstand.
Most concrete building structures are braced (nonsway) frames, with the bracing pro-vided by shear walls, stairwells, or elevator shafts that are considerably stiffer than thecolumns themselves (Fig. 10-4). Occasionally, unbraced frames are encountered near the topsof tall buildings, where the stiff elevator core may be discontinued before the top of the build-ing, or in industrial buildings where an open bay exists to accommodate a travelling crane.
Most building columns fall in the short-column category [12-1]. Exceptions occur inindustrial buildings and in buildings that have a high first-floor story for architectural orfunctional reasons. An extreme example is shown in Fig. 12-7. The left corner column hasa height of 50 times its least thickness. Some bridge piers and the decks of cable-stayedbridges fall into the slender-column category.
566 Chapter 12 Slender Columns
Fig. 12-7Bank of Brazil building,Porto Alegre, Brazil. Eachfloor extends out over thefloor below it. (Photographcourtesy of J. G. MacGregor.)
Organization of Chapter 12
The presentation of slender columns is divided into three progressively more complexparts. Slender pin-ended columns are discussed in Section 12-2. Restrained columns innonsway frames are discussed in Sections 12-3 and 12-4. These sections deal with effects and build on the material in Section 12-2. Finally, columns in sway frames arediscussed in Sections 12-5 to 12-7.
12-2 BEHAVIOR AND ANALYSIS OF PIN-ENDED COLUMNS
Lateral deflections of a slender column cause an increase in the column moments, as illus-trated in Figs. 12-1 and 12-2. These increased moments cause an increase in the deflec-tions, which in turn lead to an increase in the moments. As a result, the loadmoment lineOB in Fig. 12-2 is nonlinear. If the axial load is below the critical load, the process willconverge to a stable position. If the axial load is greater than the critical load, it will not.This is referred to as a second-order process, because it is described by a second-orderdifferential equation (Eq. 12-3).
In a first-order analysis, the equations of equilibrium are derived by assuming that thedeflections have a negligible effect on the internal forces in the members. In a second-orderanalysis, the equations of equilibrium consider the deformed shape of the structure. Instabilitycan be investigated only via a second-order analysis, because it is the loss of equilibrium of
Section 12-2 Behavior and Analysis of Pin-Ended Columns 567
the deformed structure that causes instability. However, because many engineering calcula-tions and computer programs are based on first-order analyses, methods have been derivedto modify the results of a first-order analysis to approximate the second-order effects.
Moments and Moments
Two different types of second-order moments act on the columns in a frame:
1. moments. These result from deflections, of the axis of the bent columnaway from the chord joining the ends of the column, as shown in Figs. 12-1, 12-11, and 12-13.The slenderness effects in pin-ended columns and in nonsway frames result from effects, as discussed in Sections 12-2 through 12-4.
2. moments. These result from lateral deflections, of the beamcolumnjoints from their original undeflected locations, as shown in Figs. 12-30 and 12-31. The slen-derness effects in sway frames result from moments, as discussed in Sections 12-5through 12-7.
Material Failures and Stability Failures
Loadmoment curves are plotted in Fig. 12-8 for columns of three different lengths, allloaded (as shown in Fig. 12-1) with the same end ecce