# 2015_Lecture_07_LTB_lateral_torsional_buckling - An...

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An Introduction to Instabilities Buckling Lateral Torsional Buckling by Will Laufs Civil Engineering Department Columbia University New York City
Lateral torsional buckling of beams: Example: Lateral torsional I- beam buckling lateral buckling includes lateral flexion and torsion: slide - 2 -
Torsion reminders: Elastic torsion equation (Saint-Venant theory): Parameters: - JT: quadratic moment (from geometry)- G: transversal shear elasticity module (from material) - v: Poisson’s ratioExample: slide - 3 -
Geometrical property of cross-sections: Generally, beams work mainly under flexion Cross-section are designed to resist to flexion (H beam) When other efforts become relevant, other section can be more efficient slide - 4 -
Lateral torsional buckling principle: traditional buckling problem for flange under compression could be prevented by presence of the web and the stability of the flange under tension could be prevented by adding bracing systems or lateral restraint which prevent lateral deflection slide - 5 -
Prandtl problem formulation: model: hypothesis: beam is initially straight elastic behavior uniform equally flanged I-section ends simply supported in the lateral plane (twist and lateral deflection prevented, no rotational restraint in plan = ‘fork’ support) loaded by equal and opposite end moments in the plane of the web. slide - 6 -
Prandtl equation (1899): Differential equations about weak axis: Bending: Torsion: with: bending stiffness (z - z = weak axis) bending moment resulting on torsion y St-Venant torsional stiffness warping stiffness torsion resulting on lateral deflection v (2) (1) slide - 7 -
(2) in (1) differential equation of lateral buckling: General solution (Prandtl solution): With: A1, A2, B1, B2 found through boundary conditions slide - 8 -
Solution for a simply supported beam under flexion: Boundary conditions: leads to the critical moment: slide - 9 -