COLUMN BUCKLING AND COMPRESSION MEMBER DESIGN
When a structural member is subjected to a compressive axial force, it's referred
as a compression member or a column. Compression members are found as columns in
buildings, piers in bridges, top chords of trusses. They transmit weight of
an object above it to a lower one. During this transmission, they are
compressed.
If a long slender bar is loaded, it will bend and buckle before it
yields. Due
to the sudden nature of the buckling, special attention is needed during design. This failure mode is different from the yield or fatigue and
named as instability.
Compression members are classified according to slenderness ratio and loading
type (central or eccentric loading) and analyze methods for each category is different.
Columns categorization, formulas for the determination of critical loads for different categories and loading types can be summarized as follows;
Category 
Solution with 
Long columns with central loading 
Euler column formula 
Intermediatelength columns with central loading 
J. B. Johnson formula 
Columns with eccentric loading 
Secant column formula 
Struts or short columns with eccentric loading 
Stress formulas 
SAMPLE
COLUMN 
LOADING TYPE 
CENTRAL LOADING 
ECCENTRIC LOADING 



According to design requirements, different end conditions such as fixed,
guided, pinned and free end can be used in compression members. Effect of
end conditions shall be taken into account for the determination of critical loads
for compression member.
The compression member calculation tool was developed to analyze compression members (columns).
The calculation tool finds category of the compression member according to
input parameters entered and uses relevant formula for the solution. Sample
examples for two different categories are given in "Examples" section of this
page to show the usage of calculation tool.
The formulas used for the calculations are given in the "List of Equations"
section.
Calculator:
Note 1: Use dot "." as decimal separator.
Note 2:
* If not known, use same value with yield
strength
Note 3: ** Appropriate design factor shall be selected to cover material
nonuniformity, uncertainty of service conditions, calculation and analysis
inaccuracy etc.
RESULTS 
Parameter 
Symbol 
Value 
Unit 
Effective length constant * 
C 
1.2


Radius of gyration of column 
r 



Slenderness ratio of column 
S 



Effective slenderness ratio of column 
S_{eff} 



Critical load for failure 
F_{c} 



Allowable load(includes n_{d}) 
F_{a} 


Factor of safety** 
fos 



Column Category 

Note 1:
* Suggested values per manual of steel construction.
Note 2:
** Shall be larger than design factor. Green color means safe, red color
means not safe according to input parameters.
Definitions:
Allowable stress: If a calculated maximum stress of a member for the expected service conditions is less than
a certain value and if this certain value has a proper margin against failure stress, then this value is called allowable stress. The allowable stress shall be
less than failure stress with some margin because of uncertainty of the conditions of service, nonuniformity of material, and inaccuracy of stress analysis.
In other words, allowable stress is a maximum load which can be safely placed on a structure.
Buckling:
Sudden failure of a structural member subjected to compressive stress, where the compressive stress at the point of failure is less than the
ultimate compressive strength of the material. The failure is due to the instability of the structure.
Failure stress: Loss of function stress for the design. Yield strength, proof strength,
ultimate tensile strength, critical load for instability (buckling) can be selected for failure stress but this selection depends design,
material in use and type of analysis that will performed.
Design factor (nd):The ratio of failure stress to allowable stress. The design factor is what the item is required
to withstand .The design factor is defined for an application (generally provided in advance and often set by regulatory code or policy) and
is not an actual calculation.
Eccentricity: If a load act on a member with some offset to the centroid of a member, it’s called eccentric
loading. The perpendicular distance from the line of action of the load to either principal central axis is the eccentricity with respect to that axis.
Effective length : The effective length of a column is the distance between successive points which has zero
moment.
Effective slenderness ratio: The ratio of effective length of column to the radius of gyration both with respect
to the same axis of bending. The ratio is used as a means of assessing the stability of the element. Higher slenderness ratio results lower failure loads.
Factor of Safety (Safety Factor):
The ratio of failure stress to actual/expected stress. The difference between the factor of safety (safety factor) and design factor is:
The factor of safety gives the safety margin of designed part against failure. The design factor gives the requirement value for the design.
Safety factor shall be greater than or equal to design factor.
Modulus of elasticity (Young’s
modulus): The rate of change of unit tensile or compressive stress with respect to unit tensile or compressive strain for the condition of
uniaxial stress within the proportional limit. Typical values: Aluminum: 69 GPa, Steel: 200GPa.
Neutral axis: The line of zero
fiber stress in any given section of a member subject to bending; it is the line
formed by the intersection of the neutral surface and the section.
Neutral surface: The
longitudinal surface of zero fiber stress in a member subject to bending, it
contains the neutral axis of every section.
Proportional Limit: The largest value of stress up to which a linear relation still exist between
stress and strain (Hooke’s Law).
Radius of Gyration (Area): The distance from an axis at which the area of a body may be assumed
to be concentrated and the second moment area of this configuration equal to the second moment area of the actual body about the same axis.
Stability: The structure's ability to support a given load without experiencing
a sudden change in its configuration.
Yield strength: The stress at which a material exhibits a specified permanent deformation or set.
Example: Al6061T6: 145 MPa
Supplements:
List of Equations and calculation steps:
List of equations and calculation steps for compression member design.
Examples:
Link 
Usage 
Compression of Round Column

An example about the calculation of allowable load on a compression
member. Sectional properties of the member are also calculated. 
Eccentric Compression Load On IBeam

An example about the calculation of factor of safety of a compression
member which is loaded eccentrically. Sectional properties of the member are also calculated. 
Reference: