Column Buckling

Column Properties

This section details the properties of the column material and geometry based on the provided inputs.

Inputs

The specified inputs are summarized in the table below:

The column properties are summarized in the table below. The calculations of the column properties follow the table.

The effective length factor, K, is based on the end condition:

The eccentricity ratio of this column is:

Reference Values

e	=
c	=
r	=

The transition slenderness ratio is calculated below. A column with a slenderness ratio above this value is considered long:

Reference Values

K	=
Sy	=
E	=

Results Details

This section details the column results. Refer to the column buckling reference for details further on how these results were derived.

Long Column with Central Load

Since this is a long column with a centrally applied load, the Euler formula is used to calculate the critical stress:

Reference Values

E	=
K	=
L	=
r	=

The critical force corresponding to the critical stress above is:

Pcr = σcr A =

Reference Values

σcr	=
A	=

Intermediate Column with Central Load

Since this is an intermediate column with a centrally applied load, the Johnson formula is used to calculate the critical stress:

Reference Values

Sy	=
E	=
K	=
L	=
r	=

The critical force corresponding to the critical stress above is:

Pcr = σcr A =

Reference Values

σcr	=
A	=

Column with Eccentric Load

Since this column has an eccentric load, the secant formula is used to calculate the critical force. The critical force is found by iterating the force value, P, until the calculated maximum compressive stress equals the compressive yield strength of the material (in this case, S_y = ). At a force of P = , the maximum compressive stress in the column is:

Therefore, we can conclude that the critical force is:

Pcr =

Reference Values

P	=
Sy	=
E	=
K	=
A	=
L	=
e	=
r	=
c	=

The critical stress is the average compressive stress above which the column buckles, and is calculated as:

Reference Values

Pcr	=
A	=

Critical Force Plot

The plot below shows the critical force as a function of column length. The critical force is the compressive force above which the column will buckle.

Critical Stress Plot

The plot below shows the critical stress as a function of column length. The critical stress is the value of average compressive stress above which the column will buckle, where the average stress is calculated simply as σ = P/A.

Critical Values

As the column length increases, it becomes more susceptible to buckling. The critical length is the length above which the column will buckle. The critical length, read from the plots above, is:

The critical values of force and stress were calculated previously, but are given below for reference:

Current Values

The current values of interest (based on applied loads and column geometry) are required in order to determine the factor of safety.

The applied compressive force and the column unsupported length were specified as inputs:

The average compressive stress in the column is calculated as:

Reference Values

Papp	=
A	=

Current Values for Eccentric Loading

Since the loading on this column is eccentric, there are a couple more values of interest to calculate.

The maximum deflection in the column is calculated as:

The maximum compressive stress in the column is calculated as:

Reference Values

P	=
E	=
K	=
A	=
L	=
e	=
r	=
c	=

Factor of Safety

The factor of safety is calculated as the ratio of the critical stress to the applied stress (or equivalently, the critical force to the applied force).

The Factor of Safety (FS) on stress and on applied force is: