DESIGN OF COMPRESSION MEMBERS (COLUMNS)

When a structural member is subjected to compressive axial forces, it's referred as compression member or column. Compression members are found as columns in buildings, piers in bridges, top chords of trusses. They transmit the weight of an object above it to a lower one. During this transmission, they are compressed.

Compression members are classified according to slenderness ratio and loading type (central or eccentric loading) and analyze method 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
Intermediate-length 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
Compression member (column) example Central loading of compression member Eccentric loading of compression member

According to design requirements, different end conditions such as fixed, guided, pinned and free end can be used in compression members. The effect of end condition shall be taken into account for the determination of critical load of compression member.

END CONDITIONS
Compression member end conditions - free end Compression member end conditions - fixed end Compression member end conditions - guided end Compression member end conditions - pinned end

Compression member calculation tool is developed to analyze compression members (columns). The calculation tool finds the category of the compression member according to input parameters entered and uses the 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:

INPUT PARAMETERS
Parameter Symbol Value Unit Link
Column length L  
Cross-sectional area A Info
Second moment of area I
Distance to the neutral axis c  
Eccentricity e  
Design factor** nd    
Modulus of Elasticity E  
Yield strength Sy  
Compressive yield strength* Syc
Applied Force P  
End conditions

 



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 Seff ---
Critical load for failure Fc ---
Allowable load(includes nd) Fa ---
Factor of safety** fos ---
Column Type

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: Al6061-T6: 145 MPa

Supplements:

Link Usage
Sectional Properties Calculator of Profiles Sectional properties needed for the compression/buckling analysis can be calculated with sectional properties calculator.

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 I-Beam 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:
  • Budynas.R , Nisbett.K . (2008) . Shigley's Mechanical Engineering Design . 8th edition.  McGraw-Hill
  • Beer.F.P. , Johnston.E.R. (1992). Mechanics of Materials , 2nd edition. McGraw-Hill
  • Young.W.C., (1989). ROARK’S formulas for Stress & Strain. 6th Edition. McGraw-Hill