# POST BUCKLING PHENOMENA

## Engineering Fundamentals

### INTRODUCTION

Standard buckling equations are commonly known. However, through FEM-development, analysis of post buckling behavior is possible and now elements in post buckling state can be used in construction design to the benefit of the constructions’ performance.

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###### Negative stiffness effect

A buckled leaf spring comprises a negative stiffness in the lateral direction for the range:

With

Beyond this range the buckling is ‘transforming’ into a pure s-mode bending which is reached at: . With moving back the buckling does not re-occur. In combination with manufacturing tolerances-effects, the (JPE-) working range is:

###### Buckling force

The buckling force can be considered constant after buckling and can be determined with:

###### Linear stiffness

The vertical (longitudinal) stiffness is zero. And as said, laterally it comprises a negative stiffness. The y-stiffness is similar to the transverse stiffness of a unbuckled leaf spring:

###### Stress

The maximum (Von Mises) stress is located in the ‘bending poles’ of the buckling shape which are on . For the location of the maximum stress, see the red spots in the picture. An approximation of the stress:

Thus decreases when moving in lateral direction: the buckling ‘gets relieved’. Maximum compression can be approximated with:

###### Reaction Moment

Obviously, to maintain the shape as depicted a moment at the top must be applied, this is qualified in the graph. This moment can be approximated with:

###### Rules of thumb
• is not affected by or by
• is not affected by or by
• is affected with and by
• The range of motion can be increased with or
• The stress is at the ‘bending pole’ (see graphical layout);
• movement towards the “belly” at
• movement away at