# Buckling Coefficient

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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.

When a slender structural member is subjected to compressive forces, it may buckle, which refers to a sudden, uncontrollable lateral deflection or deformation.  Buckling can lead to structural failure if not properly addressed in the design.

### Buckling coefficient formula

Fixed-Fixed (both ends are fixed)

Pinned-Pinned (both ends are hinged or pinned)

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

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

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

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

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

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

Symbol English Metric
$$K$$ = Buckling Coefficient (Fixed-fixed and Pinned-pinned) $$dimensionless$$ $$dimensionless$$
$$\lambda$$  (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)

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

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

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

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

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

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

Symbol English Metric
$$K$$ = Buckling Coefficient (Fixed-free and Pinned-free) $$dimensionless$$ $$dimensionless$$
$$\lambda$$  (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 $$in$$ $$mm$$