# Buckling Coefficient

on . Posted in Dimensionless Numbers

Buckling coefficient, abbreviated as K, also called slenderness ratio, a dimensionless number, is used in structural engineering to assess the stability of a slender structural element under axial compression that can lead to failure.  When a structure is subjected to compressive stress, buckling may occure.  Buckling is characterized by a sudden sideways deflection of a structural member.  The formula for the buckling coefficient depends on the type of end support conditions and the geometry of the column.

### Buckling coefficient formula

Fixed-Fixed (both ends are fixed)

Pinned-Pinned (both ends are hinged or pinned)

$$K = \sqrt{ \lambda \; I \;/\; k \; A_c \; l^2 }$$

$$\lambda = K^2 \; k \; A_c \; l^2 \;/\; I$$

$$I = K^2 \; k \; A_c \; l^2 \;/\; \lambda$$

$$k = \lambda \; I \;/\; K^2 \; A_c \; l^2$$

$$A_c = \lambda \; I \; k \;/\; K^2 \; l^2$$

$$l = \sqrt{ \lambda \; I \;/\; k \; A_c \; K^2 }$$

Symbol English Metric
$$K$$ = buckling coefficient (fixed-fixed and pinned-pinned) $$dimensionless$$
$$\lambda$$  (Greek symbol lambda) = elastic modulus of material $$lbf \;/\; in^2$$ $$Pa$$
$$I$$ = second moment of inertia $$in^4$$ $$mm^4$$
$$k$$ = effective length factor (which depends on the end conditions) $$in$$ $$mm$$
$$A_c$$ = area cross-section of material $$in^2$$ $$mm^2$$
$$l$$ = length of the member $$in$$ $$mm$$

### Buckling coefficient formula

Fixed-Free (one end is fixed, and the other end is free)

Pinned-Free (one end is pinned, and the other end is free)

$$K = \sqrt{ 2 \; \lambda \; I \;/\; k \; A_c \; l^2 }$$

$$\lambda = K^2 \; k \; A_c \; l^2 \;/\; 2 \; I$$

$$I = K^2 \; k \; A_c \; l^2 \;/\; 2 \; \lambda$$

$$k = K^2 \;/\; 2 \; \lambda \; A_c \; l^2$$

$$A_c = K^2 \; k \; l^2 \;/\; 2 \; \lambda \; I$$

$$l = \sqrt { k \; A_c \;/\; 2 \; \lambda \; I \; K^2 }$$

Symbol English Metric
$$K$$ = buckling coefficient (fixed-free and pinned-free) $$dimensionless$$
$$\lambda$$  (Greek symbol lambda) = elastic modulus of material $$lbf \;/\; in^2$$ $$Pa$$
$$I$$ = second moment of inertia $$in^4$$ $$mm^4$$
$$k$$ = effective length factor (which depends on the end conditions) $$in$$ $$mm$$
$$A_c$$ = area cross-section of material $$in^2$$ $$mm^2$$
$$l$$ = length of the member $$in$$ $$mm$$