Eccentric Load

A load imposed on a structural member at some point other than the centroid of the section

A compressive or tensile load which does not act through the centroid of the cross section.

Columns

The column is one of the most familiar architectonic icons. Each of us can picture this very

important space definer in our mind's eye. It can delineate, punctuate, or seperate spaces. A

quick glance out most windows allows a view of at least one of them. Columns are vertical load-

bearing elements which are normally loaded in compression. Time has brought many

manifestations of this simple element; thick, thin, long, short, spindle shaped, squat, etc. And

they have been made out of any material that has a minimal compressive strength.

The transition between the column and its loading member, usually a beam of some sort, has

often been of great interest to builders. The structural/functional problem is quite simple - the

forces collected by the beam must be transferred from the beam to the column - and yet the

solutions demonstrated a great variation. The early Egyption, Greek and Roman civilizations

turned this transition zone into an articulated flourish to draw attention to the capital. In

almost every case it was broader at the top and tapered to the column shaft. This not only

facilitated a smooth transition for the forces, but also eased the construction and even

increased the stability of of the buildings.

A column either crushes (a strength falure) or it buckles (a stability failure). Both modes of

failure must be considered for every column. The exact mode of failure is greatly dependent

upon the way in which the column's cross-sectional area is distributed with respect to its

centroid. The following simple concept must be satisfied at all times:

Stress due to loading < Resistance potential of the column

This states that the stress within the column due to all of the applied loads must be less than

the allowable stress of the material. This is a logical statement that is the essence of the

structural analysis of a column. The actual failure mechanism could be due to a combination of

two or more loads. These combinations must be carefully considered.

Eccentric Axial Load

n the case of an ideal column under an axial load, the column remains straight until the critical

load is reached. However, the load is not always applied at the centroid of the cross section, as

is assumed in Euler buckling theory. This section analyzes a simply-supported column under an

eccentric axial load.

Consider a column of length L subject to an axial force F. On one end of the column, the force F

is applied a distance e from the central column axis, as shown in the schematic below.

Balance the moments on the free-body diagram on the right requires that,

The governing equation for the column's transverse displacement w can then be written as,

where M was eliminated using Euler-Bernoulli beam theory. The above equation contains a

non-homogeneous term -Fe/EI and its general solution is,

where . . The coefficients A and B depend on the boundary conditions. For a simply

supported column the boundary conditions are,

The solution for the column's displacement is therefore,

where . .

Secant Formula

In practice, engineers are usually interested in the maximum stress rather than the

displacement curve alone. The secant formula discussed in this section derives the maximum

stress from the displacement formula obtained in the previous section.

The normal stress in the column results from both the direct axial load F and the bending

moment M resulting from the eccentricity e of the force application,

where A is the cross-section area, and I is the moment of inertia of the cross section.

The maximum stress is located at the extreme fiber on the concave side (y = c) of the middle

point (x = L/2) of the column,

where,

(obtained by applying basic trigonometric relations to the displacement formula in the previous

section). The parameter c is the distance from the centroidal axis to the extreme fiber on the

concave side of the column.

Expanding the formula for the maximum stress, we have,

The radius of gyration r is defined as . Working r into the above stress equation results

in the secant formula for maximum stress,

The secant formula indicates that in addition to the axial load F and cross-section area A, the

maximum stress also depends on the eccentricity ratio ec/r2 and the slenderness ratio L/r.

Column Failure Modes

CRUSHING

Relatively short columns are more apt to fail by the material crushing. Every building material

material can withstand a distinct amount of compressive stress before it crushes. This value has

been determined by laboratory tests and is known as the compressive strength of a material.

This strength is dependent upon the internal structure of the material and/or its components.

Steel has a very homogeneous, finely crystalled internal structure and has a relatively high

compressive strength. Hard woods are fine-grained and have a higher compressive strength

than soft-woods. Wood is unique in that it has two compressive strengths; one when loaded

parallel to the grain and another when loaded perpendicular to the grain. Why is that? When a

wood column crushes the fibers of the wood actually split apart. In every case, crushing is a

strength failure and does not depend upon the shape of the section.

BUCKLING

Relatively slender columns are more apt to fail by buckling. A column is slender when it has a

"small" cross-section compared to its effective length. Small is placed in quotes due to the fact

that the important information about the cross section is both the actual size and more

importantly, the shape of the cross-section. This is then compared to the effective length to

determine whether or not the column is slender. If it is, this means that the column will

probably fail in bending! As a column is loaded, it is likely to bend about the weak axis of the

cross-section (the one with the lowest Moment of Inertia). A column buckles when it bends

about an axis. This is a stability failure.

TYPES OF LOADING

There are generally two types of column loading: axial and eccentric. Axially loaded columns

may fail either by crushing or buckling. Eccentrically loaded columns usually fail by buckling. The

figure below illustrates the stress that a column experiences as a load, N, is applied with

increasing eccentricity. Note how the form of the stress prism changes from an even

distribution to a very uneven distribution.

The case at the left illustrates how the axial load creates a compressive stress which is evenly

distributed across the column's section. The load on each column to the right has an increasing

eccentricity. As the load moves away from the centroidal axis, it introduces a bending moment

which the column's cross-section must also resist. Thus, one can see that one side of the

column receives more compression than the other. As long as the applied load remains within

what is known as the core (the middle third) of the section, the column cross-section will only

have compressive stresses. Tesion stresses are introduced as soon as the applied load moves

out of the core. The magnitude of the bending moment that the section must also resist

increases as the eccentricity increases. The extreme case is an infinite eccentricity resulting in

the pure bending moment stress prism (like that found in a beam) that is seen in the case

furthest to the right.

Knowledge of the magnitude and distribution of the internal stress is important for the sizing of

the column. A small amount of tensile stress has little effect on a wood or steel column, but

problems could begin to occur if the column is concrete or masonry.