Elastic buckling of columns with variable section: notes

This form uses the method of stepwise approximations to solve the equation **M**=-**EJy**'' for columns undergoing elastic buckling.

**J** is taken as **J**(ξ)=**J**2ξ**n** where
**n** may typically take one of the values 1,2,3,4 and ξ is a modified abscissa as shown in forms formulae.

The following examples illustrate different values of**n**.

Beam with rectangular solid section, width varying linearly with ξ:**n**=1

Beam with rectangular hollow section and constant thickness, width varying linearly with ξ:**n**=1 (approximate)

Beam with rectangular solid section, depth varying linearly with ξ:**n**=3

Beam with rectangular hollow section and constant thickness, depth varying linearly with ξ:**n**=2 (approximate)

Truss formed with 4 slender members latticed together, width of truss varying linearly with ξ:**n**=2 (approximate)

Round solid bar, radius varying linearly with ξ:**n**=4

Round hollow bar with constant thickness, outer radius varying linearly with ξ:**n**=3 (approximate)

Round hollow bar with constant section area, outer radius varying linearly with ξ:**n**=2 (approximate)

The solution is in the form of a mixed logarithmic-polynomial approximate formula.

Numerical rounding off errors may impair the exactness of the result when the smaller and bigger moments of inertia are close each other.

Always cross check the solution with a stepped bar when the ratio of the two moments of inertia is higher than 0.8!

The following examples illustrate different values of

Beam with rectangular solid section, width varying linearly with ξ:

Beam with rectangular hollow section and constant thickness, width varying linearly with ξ:

Beam with rectangular solid section, depth varying linearly with ξ:

Beam with rectangular hollow section and constant thickness, depth varying linearly with ξ:

Truss formed with 4 slender members latticed together, width of truss varying linearly with ξ:

Round solid bar, radius varying linearly with ξ:

Round hollow bar with constant thickness, outer radius varying linearly with ξ:

Round hollow bar with constant section area, outer radius varying linearly with ξ:

The solution is in the form of a mixed logarithmic-polynomial approximate formula.

Numerical rounding off errors may impair the exactness of the result when the smaller and bigger moments of inertia are close each other.

Always cross check the solution with a stepped bar when the ratio of the two moments of inertia is higher than 0.8!