Support Conditions for Lateral-Torsional Buckling

A member's boundary conditions decisively influence the elastic critical moment for lateral-torsional buckling M_cr. The program uses a planar model with four degrees of freedom for its determination. The corresponding coefficients k_z and k_w can be defined individually for standard-compliant cross-sections. This allows you to describe the degrees of freedom available at both member ends due to the support conditions.

The effective length coefficient k_z controls the lateral displacement u_y and the rotation φ_z at the member ends. The following options are available in the list of the table column:

k_z = 1.0 restrained against lateral displacement u_y on both member ends
k_z = 0.7le restrained against displacement u_y on both member ends; restraint about z on left member end
k_z = 0.7ri restrained against displacement u_y on both member ends; restraint about z on right member end
k_z = 0.5 restrained against displacement u_y and restraint about z on both member ends
k_z = 2.0le restrained against displacement u_y and restraint about z on left member end; right member end free
k_z = 2.0ri restrained against displacement u_y and restraint about z on right member end; left member end free

Selecting Effective Length Coefficient

The warping length factor k_w controls the torsion around the member's longitudinal axis ϕ_x and the warping ω. The list offers the following options:

k_w = 1.0 restrained against rotation around x on both member ends; free to warp on both sides
k_w = 0.7le restrained against rotation around x on both ends; warping restraint on left member end
k_w = 0.7ri restrained against rotation around x on both ends; warping restraint on right member end
k_w = 0.5 torsion and warping restraint on both member ends
k_w = 2.0le restrained against rotation around x and warping ω on left member end; right member end free
k_w = 2.0ri restrained against rotation around x and warping ω on right member end; left member end free

The abbreviations "li" and "ri" refer to the left and right member ends, respectively. The abbreviation "le" always describes the support conditions at the start of the member.

Cantilever Example

A cantilever is subjected to a moment and an axial force.

Cantilever

In design case 1, the support conditions are defined as for a single-span beam with end fork conditions: k_z = 1.0 and k_w = 1.0. This results in an elastic critical moment for lateral-torsional buckling of 761.14 kNm.

Elastic Critical Moment for Single-Span Beam

The mode shape shows the lateral-torsional buckling behavior of a single-span beam.

Mode Shape for Single-Span Beam

In design case 2, the cantilever's support conditions are defined correctly: k_z = 2.0le and k_w = 2.0le. The program determines a significantly smaller critical moment of 371.72 kNm.