Engineering Fundamentals


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 u_x=\sqrt{\frac {5}{3}u_zL}

Beyond this range the buckling is ‘transforming’ into a pure s-mode bending which is reached at: x=±u_x. With moving back the buckling does not re-occur. In combination with manufacturing tolerances-effects, the (JPE-) working range is:
-\frac{1}{2}u_x<x<\frac {1}{2}u_x

Buckling force

The buckling force can be considered constant after buckling and can be determined with:
F_{buckling}=4\pi ^2\frac{EI}{L^2}

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:





The maximum (Von Mises) stress is located in the ‘bending poles’ of the buckling shape which are on \frac {1}{14}L. For the location of the maximum stress, see the red spots in the picture. An approximation of the stress:
\sigma _{max}=53\ {\cdot}\ \frac{Et}{L}\ {\cdot}\ \frac{u_z}{L}\ {\cdot}\ \sqrt{1-\left(\frac{|x|}{u_x}\right)^3}
_{(1st\ order\ estimate:\ please\ verify\ with\ FEM\ or\ consult\ JPE)}

Thus \sigma _{max} decreases when moving in lateral direction: the buckling ‘gets relieved’. Maximum compression can be approximated with:
u_{z-max}=\frac{\sigma _{allowable}L^2}{53Et}
_{(1st\ order\ estimate:\ please\ verify\ with\ FEM\ or\ consult\ JPE)}

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:
M_y=\ -3.2E\ {\cdot}\ t^3\ {\cdot}\ \frac{u_z}{L}\ {\cdot}\ \sqrt{1-\left(\frac x{0.87u_x}\right)^2}
_{(1st\ order\ estimate:\ please\ verify\ with\ FEM\ or\ consult\ JPE)}
_{(negative\ implies\ pointing\ out\ of\ the\ sheet\ with\ right-hand-rule)}

Graphical layout
Buckling phenomena - Graphical layout
Necessary forces for buckling and lateral shift
Buckling phenomena - Necessary forces for buckling and lateral shift
Rules of thumb
  • C_x is not affected by u_z or byu_x
  • F_z is not affected by or u_z by u_x
  • \sigma _{max} is affected with u_z and by u_x
  • The range of motion can be increased with \sqrt L or \sqrt{u_z}
  • The stress is at the ‘bending pole’ (see graphical layout);
    • movement towards the “belly” at \frac {1}{14}L
    • movement away at L-u_z-\frac {1}{14}L

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